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Macaulay2Doc : Index

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! -- factorial
!= -- inequality
" -- delineate a string with quotation marks
# -- length or access to elements
# BasicList -- length or cardinality
# HashTable -- length or cardinality
# Set -- length or cardinality
# String -- length or cardinality
#? -- check existence of value in a list, hash table, database, or string
% -- a binary operator, usually used for remainder and reduction
& -- a binary operator
(*) -- a unary postfix operator, used for indicating a graded object
* -- a binary operator, usually used for multiplication
** -- a binary operator, usually used for tensor product or Cartesian product
+ -- a unary or binary operator, usually used for addition
+ CC -- a unary or binary operator, usually used for addition
+ Constant -- a unary or binary operator, usually used for addition
+ QQ -- a unary or binary operator, usually used for addition
+ RR -- a unary or binary operator, usually used for addition
+ RRi -- a unary or binary operator, usually used for addition
+ ZZ -- a unary or binary operator, usually used for addition
++ -- a binary operator, usually used for direct sum
, -- the comma, used for separating entries in a list or sequence
- -- a unary or binary operator, usually used for negation or subtraction
- CC -- a unary or binary operator, usually used for negation or subtraction
- ChainComplexMap -- a unary or binary operator, usually used for negation or subtraction
- Constant -- a unary or binary operator, usually used for negation or subtraction
- Expression -- the class of all expressions
- GradedModuleMap -- a unary or binary operator, usually used for negation or subtraction
- Holder -- a unary or binary operator, usually used for negation or subtraction
- InfiniteNumber -- a unary or binary operator, usually used for negation or subtraction
- List -- negation of a vector
- Matrix -- a unary or binary operator, usually used for negation or subtraction
- Minus -- a unary or binary operator, usually used for negation or subtraction
- MutableMatrix -- a unary or binary operator, usually used for negation or subtraction
- ProjectiveHilbertPolynomial -- a unary or binary operator, usually used for negation or subtraction
- QQ -- a unary or binary operator, usually used for negation or subtraction
- RingElement -- a unary or binary operator, usually used for negation or subtraction
- RR -- a unary or binary operator, usually used for negation or subtraction
- RRi -- a unary or binary operator, usually used for negation or subtraction
- Vector -- negation of a Vector
- VirtualTally -- negation of a VirtualTally
- ZeroExpression -- a unary or binary operator, usually used for negation or subtraction
- ZZ -- a unary or binary operator, usually used for negation or subtraction
-- -- comment
-> -- make a function
. -- access to elements whose key is a symbol
.. -- a binary operator, used for sequences of consecutive items
..< -- a binary operator, used for sequences of consecutive items, not including the endpoint
.? -- check for presence of elements whose key is a symbol
/ -- a binary operator, usually used for division
// -- a binary operator, usually used for quotient
/// -- delineate a string with slashes
: -- a binary operator, uses include repetition; ideal quotients
:= -- assignment of method or new local variable
; -- expression separator
< -- less than
<- -- assignment with left side evaluated
<< -- a binary operator (file output, ...)
<< Thing -- print to a file
<= -- less than or equal
<== -- a unary and binary operator
<=== -- a unary and binary operator
<==> -- a binary operator
= -- assignment
=!= -- strict inequality
== -- equality
=== -- strict equality
===> -- a binary operator
==> -- a binary operator
=> -- produce an Option
> -- greater than
> InfiniteNumber -- greater than
> ZZ -- greater than
>= -- greater than or equal
>= InfiniteNumber -- greater than or equal
>= ZZ -- greater than or equal
>> -- a binary operator, uses include bit shifting, or attaching optional inputs to functions
? -- comparison operator
? Function -- view brief documentation of a symbol
? Keyword -- view brief documentation of a symbol
? Package -- view brief documentation of a symbol
? ScriptedFunctor -- view brief documentation of a symbol
? Symbol -- view brief documentation of a symbol
? Type -- view brief documentation of a symbol
@ -- a binary operator
@@ -- a binary operator
\ -- a binary operator
\\ -- a binary operator
^ -- a binary operator, usually used for powers
^* -- a unary postfix operator, used for indicating pullback maps
^** -- a binary operator, usually used for tensor or Cartesian power
^^ -- logical exclusive-or
_ -- a binary operator, used for subscripting and access to elements
_* -- a unary postfix operator, used for indicating pushforward maps
| -- a binary operator, often used for horizontal concatenation
|- -- a binary operator
|| -- a binary operator, often used for vertical concatenation
~ -- a unary postfix operator
4ti2
a first Macaulay2 session
about -- search the documentation
about(...,Body=>...) -- search the documentation
about(Function) -- search the documentation
about(String) -- search the documentation
about(Symbol) -- search the documentation
about(Type) -- search the documentation
abs -- absolute value function
abs(CC) -- absolute value function
abs(QQ) -- absolute value function
abs(RR) -- absolute value function
abs(RRi) -- absolute value function
abs(ZZ) -- absolute value function
accumulate -- apply a binary operator repeatedly
accumulate(Function,Thing) -- apply a binary operator repeatedly
accumulate(Function,Thing,Thing) -- apply a binary operator repeatedly
accumulate(Function,Thing,VisibleList) -- apply a binary operator repeatedly
accumulate(Function,VisibleList) -- apply a binary operator repeatedly
accumulate(VisibleList,Function) -- apply a binary operator repeatedly
accumulate(VisibleList,Thing,Function) -- apply a binary operator repeatedly
Acknowledgement -- acknowledgements listed in a documentation node
Acknowledgements
acos -- arccosine
acos(CC) -- arccosine
acos(RR) -- arccosine
acos(RRi) -- arccosine
acosh -- inverse hyperbolic cosine
acosh(Number) -- inverse hyperbolic cosine
acot -- compute the inverse cotangent
acot(Number) -- compute the inverse cotangent
acoth -- compute the hyperbolic arctangent of a number
acoth(Constant) -- compute the hyperbolic arctangent of a number
acoth(Number) -- compute the hyperbolic arctangent of a number
addCancelTask -- specify that the completion of one task triggers the cancellation of another
addCancelTask(Task,Task) -- specify that the completion of one task triggers the cancellation of another
addDependencyTask -- schedule a task, but ensure that it will not run until another task finishes
addDependencyTask(Task,Task) -- schedule a task, but ensure that it will not run until another task finishes
addEndFunction -- add an ending function
addHook -- add a hook function to an object for later processing
addHook(...,Strategy=>...) -- add a hook function to an object for later processing
addHook(MutableHashTable,Thing,Function) -- add a hook function to an object for later processing
addHook(Sequence,Function) -- add a hook function to an object for later processing
addHook(Symbol,Function) -- add a hook function to an object for later processing
AdditionalPaths -- list of non-standard paths to search for a program
addStartFunction -- add a startup function
addStartTask -- schedule a task upon completion of another
addStartTask(Task,Task) -- schedule a task upon completion of another
Adjacent -- the class of all adjacent expression pairs
adjoint -- an adjoint map
adjoint' -- an adjoint map
adjoint'(Matrix,Module,Module) -- an adjoint map
adjoint(Matrix,Module,Module) -- an adjoint map
AffineVariety -- the class of all affine varieties
AffineVariety * AffineVariety -- a binary operator, usually used for multiplication
AffineVariety ** Ring -- a binary operator, usually used for tensor product or Cartesian product
AfterEval -- top level method applied after evaluation
AfterNoPrint -- top level method applied after not printing
AfterPrint -- top level method applied after printing
agm -- arithmetic-geometric mean
agm(CC,CC) -- arithmetic-geometric mean
agm(RR,RR) -- arithmetic-geometric mean
alarm -- set an alarm
Algorithm -- an optional argument
Alignment -- an optional argument
all -- whether all elements satisfy a specified condition
all(BasicList,BasicList,Function) -- whether all corresponding elements of a pair of lists satisfy a condition
all(BasicList,Function) -- whether all elements of a list satisfy a specified condition
all(HashTable,Function) -- whether all key/value pairs in a hash table satisfy a specified condition
all(ZZ,Function) -- whether all elements of a list satisfy a specified condition
allowableThreads -- the current maximum number of simultaneously running tasks
ambient -- ambient free module of a subquotient, or ambient ring
ambient(GaloisField) -- corresponding quotient ring
ambient(Matrix)
ambient(Module) -- ambient free module
ambient(PolynomialRing) -- ambient polynomial ring
ambient(QuotientRing) -- ambient polynomial ring
ambient(Ring) -- ambient polynomial ring
an example of a package
ancestor -- whether one type is an ancestor of another
ancestor(Type,Type) -- whether one type is an ancestor of another
ancestors -- the ancestors of something
and -- conjunction
AngleBarList -- the class of lists delimited by <| ... |>
antipode -- antipode for skew commuting polynomial rings
antipode(RingElement) -- antipode for skew commuting polynomial rings
any -- whether any elements satisfy a specified condition
any(BasicList,BasicList,Function) -- whether any corresponding elements of a pair of lists satisfy a condition
any(BasicList,Function) -- whether any elements of a list satisfy a specified condition
any(HashTable,Function) -- whether all key/value pairs in a hash table satisfy a specified condition
any(ZZ,Function) -- whether any elements of a list satisfy a specified condition
append -- add an element to the end of a list
append(BasicList,Thing) -- add an element to the end of a list
application directory -- the path to the user's application directory
applicationDirectory -- the path to the user's application directory
applicationDirectorySuffix -- suffix that determines the user's application directory
apply -- apply a function to each element
apply(BasicList,BasicList,Function) -- apply a function to pairs of elements, one from each list
apply(BasicList,Function) -- apply a function to each element of a list
apply(BasicList,String,Function) -- apply a function to pairs of elements, one from each list
apply(String,BasicList,Function) -- apply a function to pairs of elements, one from each list
apply(String,Function) -- apply a function to each element of a list
apply(String,String,Function) -- apply a function to pairs of elements, one from each list
apply(Thing,Function) -- apply a function to an object with an iterator
apply(ZZ,Function) -- apply a function to {0,..., n-1}
applyKeys -- apply a function to each key in a hash table
applyKeys(HashTable,Function) -- apply a function to each key in a hash table
applyKeys(HashTable,Function,Function) -- apply a function to each key in a hash table
applyPairs -- apply a function to each pair in a hash table
applyPairs(HashTable,Function) -- apply a function to each pair in a hash table
applyTable -- apply a function to each element of a table
applyValues -- apply a function to each value in a hash table
applyValues(HashTable,Function) -- apply a function to each value in a hash table
apropos -- symbols matching a pattern
apropos(String) -- symbols matching a pattern
argument -- specify the function in a scripted functor for an argument
Array -- the class of all arrays -- [...]
Array | Array -- join lists, sequences or arrays
Ascending -- specify ascending order
ascii -- ASCII character conversion
ascii(List) -- ASCII character conversion
ascii(String) -- ASCII character conversion
asin -- arcsine
asin(CC) -- arcsine
asin(RR) -- arcsine
asin(RRi) -- arcsine
asinh -- inverse hyperbolic sine
asinh(Number) -- inverse hyperbolic sine
assert -- check whether something is true, raise an error if not
assert(Expression) -- check whether something is true, raise an error if not
assert(Thing) -- check whether something is true, raise an error if not
assigning values
associative algebras
AssociativeExpression -- the class of associative expressions
atan -- compute the arctangent of a number
atan(CC) -- compute the arctangent of a number
atan(RR) -- compute the arctangent of a number
atan(RRi) -- compute the arctangent of a number
atan2 -- compute an angle of a certain triangle
atan2(RR,RR) -- compute an angle of a certain triangle
atan2(RRi,RRi) -- compute an angle of a certain triangle
atanh -- compute the hyperbolic arctangent of a number
atanh(Constant) -- compute the hyperbolic arctangent of a number
atanh(Number) -- compute the hyperbolic arctangent of a number
atEndOfFile -- test for end of file
atEndOfFile(File) -- test for end of file
Authors -- an optional argument
authors of Macaulay2 packages
autoload -- arrange for a function to be loaded automatically
autoload(Symbol,String) -- arrange for a function to be loaded automatically
AuxiliaryFiles -- an optional argument
backtrace -- whether a backtrace is displayed following an error message
Bag -- the class of all bags
Bag ? Bag -- the class of all bags
Bareiss
baseFilename -- the base part of a filename or path
BaseFunction -- the basename of a function in a documentation node
baseName -- the base name of a generator
baseName(Holder) -- the base name of a generator
baseName(IndexedVariable) -- the base name of a generator
baseName(IndexedVariableTable) -- the base name of a generator
baseName(RingElement) -- the base name of a generator
baseName(Subscript) -- the base name of a generator
baseName(Symbol) -- the base name of a generator
baseName(Thing) -- the base name of a generator
baseRing -- produce the ring from which a ring was formed
baseRing(Ring) -- produce the ring from which a ring was formed
baseRings -- store the list of base rings of a ring
BaseRow -- an optional argument
basic arithmetic of matrices
basic commutative algebra
basic construction, source and target of a ring map
basic rings of numbers
BasicList -- the class of all basic lists
BasicList # ZZ -- get value from list, hash table, database, or string
BasicList #? ZZ -- check existence of value in a list, hash table, database, or string
basis -- basis or generating set of all or part of a ring, ideal or module
basis(...,Degree=>...) -- basis or generating set of all or part of a ring, ideal or module
basis(...,Limit=>...) -- basis or generating set of all or part of a ring, ideal or module
basis(...,SourceRing=>...) -- basis or generating set of all or part of a ring, ideal or module
basis(...,Strategy=>...) -- basis or generating set of all or part of a ring, ideal or module
basis(...,Truncate=>...) -- basis or generating set of all or part of a ring, ideal or module
basis(...,Variables=>...) -- basis or generating set of all or part of a ring, ideal or module
basis(Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,InfiniteNumber,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,InfiniteNumber,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,InfiniteNumber,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,InfiniteNumber,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,List,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,List,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,List,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,List,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,ZZ,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,ZZ,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,ZZ,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(InfiniteNumber,ZZ,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(List,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(List,InfiniteNumber,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(List,InfiniteNumber,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(List,InfiniteNumber,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(List,InfiniteNumber,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(List,List,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(List,List,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(List,List,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(List,List,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(List,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(List,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(List,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(List,ZZ,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(List,ZZ,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(List,ZZ,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(List,ZZ,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(Module) -- basis or generating set of all or part of a ring, ideal or module
basis(Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,InfiniteNumber,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,InfiniteNumber,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,InfiniteNumber,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,InfiniteNumber,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,List,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,List,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,List,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,List,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,Ring) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,ZZ,Ideal) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,ZZ,Matrix) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,ZZ,Module) -- basis or generating set of all or part of a ring, ideal or module
basis(ZZ,ZZ,Ring) -- basis or generating set of all or part of a ring, ideal or module
BasisElementLimit -- an optional argument
Bayer -- value of an optional argument
BeforePrint -- top level method applied before printing results
beginDocumentation -- start documentation section of a package
benchmark -- accurate timing of execution
BesselJ -- Bessel function of the first kind
BesselJ(ZZ,Constant) -- Bessel function of the first kind
BesselJ(ZZ,Number) -- Bessel function of the first kind
BesselY -- Bessel function of the second kind
BesselY(ZZ,Constant) -- Bessel function of the second kind
BesselY(ZZ,Number) -- Bessel function of the second kind
Beta -- Beta function
betti -- display or modify a Betti diagram
betti(...,Minimize=>...) -- minimal betti numbers of a non-minimal free resolution
betti(...,Weights=>...) -- view and set the weight vector of a Betti diagram
betti(BettiTally) -- view and set the weight vector of a Betti diagram
betti(ChainComplex) -- display or modify a Betti diagram
betti(CoherentSheaf) -- display or modify a Betti diagram
betti(GradedModule) -- display or modify a Betti diagram
betti(GroebnerBasis) -- display or modify a Betti diagram
betti(Ideal) -- display or modify a Betti diagram
betti(Matrix) -- display or modify a Betti diagram
betti(Module) -- display or modify a Betti diagram
betti(MonomialIdeal) -- display or modify a Betti diagram
BettiTally -- the class of all Betti tallies
BettiTally ** BettiTally -- the class of all Betti tallies
BettiTally ++ BettiTally -- the class of all Betti tallies
BettiTally == BettiTally -- equality
BettiTally Array -- the class of all Betti tallies
BettiTally ZZ -- the class of all Betti tallies
between -- insert something between elements of a list
between(Thing,VisibleList) -- insert something between elements of a list
Binary -- a hashtable with information about Macaulay2 operators
binary methods
BinaryOperation -- the class of all binary operation expressions
Binomial -- value of an optional argument
binomial -- binomial coefficient
binomial(Constant,Constant) -- binomial coefficient
binomial(Constant,Number) -- binomial coefficient
binomial(Number,Constant) -- binomial coefficient
binomial(Number,Number) -- binomial coefficient
binomial(RingElement,ZZ) -- binomial coefficient
BLAS
blockMatrixForm -- global flag for block printing
Body -- an optional argument
Boolean -- the class of Boolean values
Boolean == Boolean -- equality
Boolean >> Function -- attaching options to a function
Boolean xor Boolean -- exclusive disjunction
Boost -- a collection of C++ source libraries
borel -- make a Borel fixed submodule
borel(Matrix) -- make a Borel fixed submodule
borel(MonomialIdeal) -- make a Borel fixed submodule
Boxes -- an optional argument
break -- break from a loop
cache -- a key under which to store cache tables
CacheExampleOutput -- an optional argument
CacheFunction -- the class of cache functions
CacheTable -- hash tables for caching
cacheValue -- cache values of functions in their arguments
CallLimit -- trace a function each time it's run
cancelTask -- stop a task
cancelTask(Task) -- stop a task
capture -- evaluate Macaulay2 code and capture the output
capture(...,PackageExports=>...) -- evaluate Macaulay2 code and capture the output
capture(...,UserMode=>...) -- evaluate Macaulay2 code and capture the output
capture(List) -- evaluate Macaulay2 code and capture the output
capture(Net) -- evaluate Macaulay2 code and capture the output
capture(String) -- evaluate Macaulay2 code and capture the output
CatalanConstant -- Catalan's constant
catch -- catch a thrown exception
Caveat -- warnings or limitations listed in a documentation node
CC -- the class of all complex numbers
CC % CC -- a binary operator, usually used for remainder and reduction
CC % QQ -- a binary operator, usually used for remainder and reduction
CC % RR -- a binary operator, usually used for remainder and reduction
CC % ZZ -- a binary operator, usually used for remainder and reduction
CC * CC -- a binary operator, usually used for multiplication
CC * QQ -- a binary operator, usually used for multiplication
CC * RR -- a binary operator, usually used for multiplication
CC * ZZ -- a binary operator, usually used for multiplication
CC + CC -- a unary or binary operator, usually used for addition
CC + QQ -- a unary or binary operator, usually used for addition
CC + RR -- a unary or binary operator, usually used for addition
CC + ZZ -- a unary or binary operator, usually used for addition
CC - CC -- a unary or binary operator, usually used for negation or subtraction
CC - QQ -- a unary or binary operator, usually used for negation or subtraction
CC - RR -- a unary or binary operator, usually used for negation or subtraction
CC - ZZ -- a unary or binary operator, usually used for negation or subtraction
CC / CC -- a binary operator, usually used for division
CC / QQ -- a binary operator, usually used for division
CC / RR -- a binary operator, usually used for division
CC / ZZ -- a binary operator, usually used for division
CC // CC -- a binary operator, usually used for quotient
CC // QQ -- a binary operator, usually used for quotient
CC // RR -- a binary operator, usually used for quotient
CC // ZZ -- a binary operator, usually used for quotient
CC << ZZ
CC == CC -- equality
CC == QQ -- equality
CC == RR -- equality
CC == ZZ -- equality
CC >> ZZ
CC ^ ZZ -- a binary operator, usually used for powers
CC' -- the parent class of all rings of complex numbers
cdd+
ceiling -- ceiling function
ceiling(Constant) -- ceiling function
ceiling(Number) -- ceiling function
Center -- value of an optional argument
centerString -- center a string or net
Certification -- an optional argument
chain complexes
ChainComplex -- the class of all chain complexes
chainComplex -- make a chain complex
ChainComplex ** ChainComplex -- tensor product
ChainComplex ** ChainComplexMap -- tensor product
ChainComplex ** GradedModule -- tensor product
ChainComplex ** Module -- tensor product
ChainComplex ** Ring -- a binary operator, usually used for tensor product or Cartesian product
ChainComplex ++ ChainComplex -- direct sum
ChainComplex == ChainComplex -- equality
ChainComplex == ZZ -- equality
ChainComplex ^ Array -- projection onto summand
ChainComplex ^ ZZ -- access member, cohomological degree
ChainComplex _ Array -- inclusion from summand
ChainComplex _ ZZ -- component
ChainComplex _ ZZ = Thing -- install component of chain complex
ChainComplex Array -- degree shift
chainComplex(GradedModule) -- make a chain complex from a graded module
chainComplex(List) -- make a chain complex
chainComplex(Matrix) -- make a small chain complex
chainComplex(Ring) -- make an empty chain complex over a ring
chainComplex(Sequence) -- make a chain complex
ChainComplexMap -- the class of all maps between chain complexes
ChainComplexMap * ChainComplexMap -- a binary operator, usually used for multiplication
ChainComplexMap ** ChainComplex -- tensor product
ChainComplexMap ** ChainComplexMap -- tensor product
ChainComplexMap ** Module -- tensor product
ChainComplexMap + ChainComplexMap -- a unary or binary operator, usually used for addition
ChainComplexMap + RingElement -- a unary or binary operator, usually used for addition
ChainComplexMap + ZZ -- a unary or binary operator, usually used for addition
ChainComplexMap ++ ChainComplexMap -- direct sum
ChainComplexMap - ChainComplexMap -- a unary or binary operator, usually used for negation or subtraction
ChainComplexMap - RingElement -- a unary or binary operator, usually used for negation or subtraction
ChainComplexMap == ChainComplexMap -- equality
ChainComplexMap == RingElement -- equality
ChainComplexMap == ZZ -- equality
ChainComplexMap ^ Array -- component of map corresponding to summand of target
ChainComplexMap ^ ZZ -- iterated composition
ChainComplexMap _ Array -- component of map corresponding to summand of source
ChainComplexMap _ ZZ -- component map
ChainComplexMap _ ZZ = Thing -- install component of chain complex map
ChainComplexMap Array -- degree shift
changeBase -- change integer base
changeBase(String,ZZ) -- change integer base
changeBase(String,ZZ,ZZ) -- change integer base
changeBase(ZZ,ZZ) -- change integer base
ChangeMatrix -- an optional argument
changes made for the next release
changes to Macaulay2, by version
changes, 1.0 and 1.1
changes, 1.2
changes, 1.3
changes, 1.3.1
changes, 1.4
changes, 1.5
changes, 1.6
changes, 1.7
changes, 1.8
changes, 1.8.1
changes, 1.8.2
changes, 1.9
changes, 1.9.1
changes, 1.9.2
changes, 1.10
changes, 1.11
changes, 1.12
changes, 1.13
changes, 1.14
changes, 1.15
changes, 1.16
changes, 1.17
changes, 1.18
changes, 1.19
changes, 1.19.1
changes, 1.20
changes, 1.21
char -- computes the characteristic of the ring or field
char(AffineVariety) -- computes the characteristic of the ring or field
char(InexactField) -- computes the characteristic of the ring or field
char(ProjectiveVariety) -- computes the characteristic of the ring or field
char(Ring) -- computes the characteristic of the ring or field
characters -- get characters from a string
check -- perform tests of a package
check(...,UserMode=>...) -- perform tests of a package
check(...,Verbose=>...) -- perform tests of a package
check(List,Package) -- perform tests of a package
check(List,String) -- perform tests of a package
check(Package) -- perform tests of a package
check(String) -- perform tests of a package
check(ZZ,Package) -- perform tests of a package
check(ZZ,String) -- perform tests of a package
CheckDocumentation -- an optional argument
checking your Macaulay2 installation -- and bookmarking the Macaulay2 documentation
chi
class -- class of an object
clean -- Set to zero elements that are approximately zero
clean(RR,Matrix) -- Set to zero elements that are approximately zero
clean(RR,MutableMatrix) -- Set to zero elements that are approximately zero
clean(RR,Number) -- Set to zero elements that are approximately zero
clean(RR,RingElement) -- Set to zero elements that are approximately zero
clearAll -- forget everything
clearEcho -- turn off echoing
clearOutput -- forget output values
close -- close a file
closeIn -- close an input file
closeOut -- close an output file
ClosestFit -- an optional argument
code -- display source code
code(Command) -- display source code
code(Function) -- display source code
code(List) -- display source code
code(Pseudocode) -- display source code
code(Sequence) -- display source code
code(Symbol) -- display source code
code(TestInput) -- locate a package's tests
code(ZZ) -- display source code
codim -- compute the codimension
codim(BettiTally) -- the class of all Betti tallies
codim(CoherentSheaf) -- codimension of the support of a coherent sheaf on a projective variety
codim(CoherentSheaf,Generic=>...)
codim(Ideal) -- compute the codimension
codim(Ideal,Generic=>...)
codim(Module) -- codimension of the support of a module
codim(Module,Generic=>...)
codim(MonomialIdeal) -- compute the codimension
codim(PolynomialRing) -- compute the codimension
codim(PolynomialRing,Generic=>...)
codim(ProjectiveVariety) -- codimension of the projective variety
codim(ProjectiveVariety,Generic=>...)
codim(QuotientRing) -- compute the codimension
codim(QuotientRing,Generic=>...)
CodimensionLimit -- an optional argument
coefficient -- coefficient of a monomial
CoefficientRing -- an optional argument
coefficientRing -- get the coefficient ring
coefficientRing(Ring) -- get the coefficient ring
coefficients -- monomials and their coefficients
coefficients(...,Monomials=>...) -- monomials and their coefficients
coefficients(...,Variables=>...) -- monomials and their coefficients
coefficients(Matrix) -- monomials and their coefficients
coefficients(RingElement) -- monomials and their coefficients
Cofactor
coherent sheaves
CoherentSheaf -- the class of all coherent sheaves
CoherentSheaf (*) -- sum of twists
CoherentSheaf ** CoherentSheaf -- tensor produce of coherent sheaves
CoherentSheaf ++ CoherentSheaf -- direct sum of coherent sheaves
CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
CoherentSheaf / Ideal -- quotient of coherent sheaves
CoherentSheaf ^ ZZ -- direct sum
CoherentSheaf ^** ZZ -- tensor power
CoherentSheaf LowerBound -- the class of lower bound objects
CoherentSheaf ZZ -- canonical twist of a coherent sheaf
cohomCalg
cohomology -- general cohomology functor
coimage -- coimage of a map
coimage(ChainComplexMap) -- coimage of a map
coimage(GradedModuleMap) -- coimage of a map
coimage(Matrix) -- coimage of a map
coimage(RingMap) -- coimage of a map
cokernel -- cokernel of a map of modules, graded modules, or chaincomplexes
cokernel(ChainComplexMap) -- cokernel of a map of modules, graded modules, or chaincomplexes
cokernel(GradedModuleMap) -- cokernel of a map of modules, graded modules, or chaincomplexes
cokernel(Matrix) -- cokernel of a map of modules, graded modules, or chaincomplexes
cokernel(RingElement) -- cokernel of a map of modules, graded modules, or chaincomplexes
collectGarbage -- collect the garbage in memory
columnAdd -- add a multiple of one column to another
columnAdd(MutableMatrix,ZZ,Number,ZZ) -- add a multiple of one column to another
columnAdd(MutableMatrix,ZZ,RingElement,ZZ) -- add a multiple of one column to another
columnate -- arrange strings in columns
columnMult -- multiply a column by a ring element
columnMult(MutableMatrix,ZZ,Number) -- multiply a column by a ring element
columnMult(MutableMatrix,ZZ,RingElement) -- multiply a column by a ring element
columnPermute -- permute some columns
columnPermute(MutableMatrix,ZZ,List) -- permute some columns
columnRankProfile -- find the column rank profile of a mutable matrix
columnRankProfile(MutableMatrix) -- find the column rank profile of a mutable matrix
columnSwap -- interchange columns
columnSwap(MutableMatrix,ZZ,ZZ) -- interchange columns
combinatorial functions
combine -- combine hash tables
Command -- the class of all commands
Command \ String -- apply a function to elements of a list
Command \ VirtualTally -- apply a function to elements of a list
Command \ VisibleList -- apply a function to elements of a list
Command \\ Thing -- apply a function
Command Thing -- the class of all commands
commandInterpreter -- the top level command interpreter
commandLine -- the command line arguments
commonest -- the most common elements of a list or tally
commonest(Set) -- the most common elements of a list or tally
commonest(Tally) -- the most common elements of a list or tally
commonest(VisibleList) -- the most common elements of a list or tally
commonRing -- find smallest containing ring
commonRing(List) -- find smallest containing ring
communicating with programs
comodule -- submodule to quotient module
comodule(Ideal) -- submodule to quotient module
comodule(Module) -- submodule to quotient module
compactMatrixForm -- global flag for compact printing
CompiledFunction -- the class of all compiled functions
CompiledFunctionBody -- the class of all compiled function bodies
CompiledFunctionClosure -- the class of all compiled function closures
Complement -- a Strategy option value
complement -- find the minimal generators for cokernel of a matrix (low level form)
complement(Matrix) -- find the minimal generators for cokernel of a matrix (low level form)
complete
complete(ChainComplex) -- complete the internal parts
complete(ChainComplexMap)
complete(GradedModule)
CompleteIntersection -- value of an optional argument
ComplexField -- the class of all complex fields
components -- list the components of a direct sum
components(ChainComplex)
components(ChainComplexMap) -- list the components of a direct sum
components(GradedModule) -- list the components of a direct sum
components(GradedModuleMap) -- list the components of a direct sum
components(Matrix) -- list the components of a direct sum
components(Module) -- list the components of a direct sum
compose -- composition as a pairing on Hom-modules
compose(Module,Module,Module) -- composition as a pairing on Hom-modules
compositions -- list the compositions of an integer
compositions(ZZ,ZZ) -- list the compositions of an integer
compress -- extract nonzero columns from a matrix
compress(Matrix) -- extract nonzero columns from a matrix
computing Groebner bases
computing resolutions
computing syzygies
concatenate -- join strings
concatenate(BasicList) -- join strings
concatenate(Nothing) -- join strings
concatenate(String) -- join strings
concatenate(Symbol) -- join strings
concatenate(ZZ) -- join strings
concatenating matrices
conditional execution -- if-then-else statements
cone -- mapping cone of a chain map
cone(ChainComplexMap) -- mapping cone of a chain map
Configuration -- an optional argument
conjugate -- complex conjugate
conjugate(CC) -- complex conjugate
conjugate(Number) -- complex conjugate
conjugate(Partition) -- conjugate a partition
connectionCount -- the number of connections
Consequences -- side-effects of a function in a documentation node
Constant
Constant ! -- factorial
Constant * Constant -- a binary operator, usually used for multiplication
Constant * InexactNumber -- a binary operator, usually used for multiplication
Constant * Number -- a binary operator, usually used for multiplication
Constant * RingElement
Constant + Constant -- a unary or binary operator, usually used for addition
Constant + InexactNumber -- a unary or binary operator, usually used for addition
Constant + Number -- a unary or binary operator, usually used for addition
Constant + RingElement
Constant - Constant -- a unary or binary operator, usually used for negation or subtraction
Constant - InexactNumber -- a unary or binary operator, usually used for negation or subtraction
Constant - Number -- a unary or binary operator, usually used for negation or subtraction
Constant - RingElement
Constant / Constant
Constant / InexactNumber
Constant / Number
Constant / RingElement
Constant == Constant
Constant == InexactNumber
Constant ^ Constant
Constant ^ InexactNumber
Constant ^ Number
Constant ^ Ring -- lift to another ring
Constant ^ RingFamily -- lift to another ring
Constants -- an optional argument
constructing maps between modules
content -- the content of a polynomial
content(RingElement) -- the content of a polynomial
content(RingElement,RingElement) -- the content of a polynomial
continue -- continue with the next iteration of a loop
contract -- contract one matrix by another
contract' -- contract a matrix by a matrix, the dual notion
contract'(Matrix,Matrix) -- contract a matrix by a matrix, the dual notion
contract(Matrix,Matrix) -- contract a matrix by a matrix
contract(Matrix,Number) -- contract a matrix by a matrix
contract(Matrix,RingElement) -- contract a matrix by a matrix
contract(Matrix,Vector) -- contract a matrix by a matrix
contract(Number,Matrix) -- contract a matrix by a matrix
contract(Number,Number) -- contract a matrix by a matrix
contract(Number,RingElement) -- contract a matrix by a matrix
contract(Number,Vector) -- contract a matrix by a matrix
contract(RingElement,Matrix) -- contract a matrix by a matrix
contract(RingElement,Number) -- contract a matrix by a matrix
contract(RingElement,RingElement) -- contract a matrix by a matrix
contract(RingElement,Vector) -- contract a matrix by a matrix
contract(Vector,Matrix) -- contract a matrix by a matrix
contract(Vector,Number) -- contract a matrix by a matrix
contract(Vector,RingElement) -- contract a matrix by a matrix
contract(Vector,Vector) -- contract a matrix by a matrix
Contributors -- non-author contributors listed in a documentation node
conventions for documentation
copy -- copy an object
copyDirectory
copyDirectory(...,Exclude=>...)
copyDirectory(...,FollowLinks=>...)
copyDirectory(...,UpdateOnly=>...)
copyDirectory(...,Verbose=>...)
copyDirectory(String,String)
copyFile
copyFile(...,UpdateOnly=>...)
copyFile(...,Verbose=>...)
copyFile(String,String)
COPYING-GPL-2 -- GNU General Public License, version 2
COPYING-GPL-3 -- GNU General Public License, version 3
copyright -- a string containing the copyright notice for Macaulay2
Copyright and license
Core -- the core part of Macaulay2
cos -- compute the cosine
cos(CC) -- compute the cosine
cos(RR) -- compute the cosine
cos(RRi) -- compute the cosine
cosh -- compute the hyperbolic cosine
cosh(CC) -- compute the hyperbolic cosine
cosh(RR) -- compute the hyperbolic cosine
cosh(RRi) -- compute the hyperbolic cosine
cot -- cotangent
cot(CC) -- cotangent
cot(RR) -- cotangent
cot(RRi) -- cotangent
cotangentSheaf -- cotangent sheaf of a projective variety
cotangentSheaf(...,Minimize=>...) -- cotangent sheaf of a projective variety
cotangentSheaf(ProjectiveVariety) -- cotangent sheaf of a projective variety
cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety
coth -- hyperbolic cotangent
coth(CC) -- hyperbolic cotangent
coth(RR) -- hyperbolic cotangent
coth(RRi) -- hyperbolic cotangent
cover -- get the covering free module
cover(CoherentSheaf) -- get the covering free module
cover(GradedModule) -- get the covering free module
cover(Matrix) -- get the covering free module
cover(Module) -- get the covering free module
coverMap -- the surjective map from a free module to a module corresponding to the generators
coverMap(Module) -- the surjective map from a free module to a module corresponding to the generators
cpuTime -- seconds of cpu time used since Macaulay2 began
createTask -- create a task
createTask(Function) -- create a task
createTask(Function,Thing) -- create a task
creating a package
creating an ideal
creating and writing files
csc -- cosecant
csc(CC) -- cosecant
csc(RR) -- cosecant
csc(RRi) -- cosecant
csch -- hyperbolic cosecant
csch(CC) -- hyperbolic cosecant
csch(RR) -- hyperbolic cosecant
csch(RRi) -- hyperbolic cosecant
csdp
current -- the pseudocode that produced an error
currentColumnNumber -- current column number of the current input file
currentDirectory -- current working directory
currentFileDirectory -- the directory containing the current input file
currentFileName -- the current source file
currentLayout -- relative locations of Macaulay2 files
currentPackage -- the current package
currentPosition -- the current position in the current source file
currentRowNumber -- current row number of the current input file
currentString -- evaluate a string
currentTime -- get the current time
Daniel R. Grayson
Database -- the class of all database files
Database # String -- get value from list, hash table, database, or string
Database #? String -- check existence of value in a list, hash table, database, or string
Date -- an optional argument
David Eisenbud
dd -- differential in a chain complex
debug -- open the private dictionary of a package
debug(Package) -- open the private dictionary of a package
debugError -- a function to debug
debugging
DebuggingMode -- an optional argument
debuggingMode -- whether to enter the debugger when an error occurs
debugLevel -- current level debugging
Decompose -- value of an optional argument
decompose
deepSplice -- remove all levels of subsequences from a sequence or list
deepSplice(BasicList) -- remove all levels of subsequences from a sequence or list
Default -- value of an optional argument
default -- default member of a family
default(InexactFieldFamily) -- default member of a family
defaultPrecision -- default precision of numbers
definition of product (block) orders
Degree -- specify the degree of a map
degree
degree(BettiTally) -- the class of all Betti tallies
degree(Ideal)
degree(Module)
degree(ProjectiveHilbertPolynomial)
degree(ProjectiveVariety)
degree(Ring)
degree(RingElement)
degree(RingElement,RingElement) -- degree with respect to a variable
degree(Vector)
DegreeGroup -- an optional argument
degreeLength -- the length of the degree vector
degreeLength(Monoid) -- the length of the degree vector
degreeLength(Ring) -- the length of the degree vector
DegreeLift -- an optional argument
DegreeLimit -- an optional argument
DegreeMap -- an optional argument
DegreeOrder -- sort primarily by degree
DegreeRank -- an optional argument
Degrees -- an optional argument
degrees -- degrees of generators
degrees(CoherentSheaf) -- degrees of generators
degrees(FractionField) -- degrees of generators
degrees(Ideal) -- degrees of generators
degrees(Matrix) -- degrees of target and source
degrees(Module) -- degrees of generators
degrees(Monoid) -- degrees of generators
degrees(QuotientRing) -- degrees of generators
degrees(Ring) -- degrees of generators
degreesMonoid -- the ring or monoid of degrees
degreesMonoid(List) -- the ring or monoid of degrees
degreesMonoid(Monoid) -- the ring or monoid of degrees
degreesMonoid(Ring) -- the ring or monoid of degrees
degreesMonoid(ZZ) -- the ring or monoid of degrees
degreesRing -- the ring or monoid of degrees
degreesRing(List) -- the ring or monoid of degrees
degreesRing(Monoid) -- the ring or monoid of degrees
degreesRing(Ring) -- the ring or monoid of degrees
degreesRing(ZZ) -- the ring or monoid of degrees
delete -- delete some elements of a list
delete(Thing,BasicList) -- delete some elements of a list
demark -- insert a string between elements of a list of strings
denominator -- denominator of a fraction
denominator(Divide) -- denominator of a fraction
Dense -- an optional argument
Density -- an optional argument
depth -- depth of a net
depth(Net) -- depth of a net
depth(String) -- depth of a net
Descending -- specify descending order
Descent
Describe -- the class of the output of describe
describe -- real description
describe(AffineVariety) -- real description
describe(CoherentSheaf) -- real description
describe(FractionField) -- real description
describe(GaloisField) -- real description
describe(Matrix) -- real description
describe(Module) -- real description
describe(Monoid) -- real description
describe(PolynomialRing) -- real description
describe(ProjectiveVariety) -- real description
describe(QuotientRing) -- real description
describe(RingMap) -- real description
describe(Thing) -- real description
determinant -- determinant of a matrix
determinant(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
determinant(Matrix) -- determinant of a matrix
determinant(MutableMatrix) -- determinant of a matrix
determinants and minors
Developer's Corner
diagonalMatrix -- make a diagonal matrix
diagonalMatrix(List) -- make a diagonal matrix from a list
diagonalMatrix(Matrix) -- make a diagonal matrix from entries of a matrix
diagonalMatrix(Ring,List) -- make a diagonal matrix from a list
diagonalMatrix(Ring,ZZ,ZZ,List) -- make a diagonal matrix from a list
diagonalMatrix(RingFamily,List) -- make a diagonal matrix from a list
diagonalMatrix(RingFamily,ZZ,ZZ,List) -- make a diagonal matrix from a list
diagonalMatrix(ZZ,ZZ,List) -- make a diagonal matrix from a list
diameter -- diameter of an interval
diameter(RRi) -- diameter of an interval
Dictionary -- the class of all dictionaries
dictionary -- determine the dictionary to which a symbol belongs
dictionary(Keyword) -- determine the dictionary to which a symbol belongs
dictionary(Symbol) -- determine the dictionary to which a symbol belongs
dictionary(Thing) -- determine the dictionary to which a symbol belongs
dictionaryPath
diff -- differentiate or take difference
diff and contract
diff' -- differentiate a matrix by a matrix, the dual notion
diff'(Matrix,Matrix) -- differentiate a matrix by a matrix, the dual notion
diff(Matrix,Matrix) -- differentiate a matrix by a matrix
diff(Matrix,RingElement) -- differentiation
diff(Matrix,Vector) -- differentiate a matrix by a matrix
diff(ProjectiveHilbertPolynomial)
diff(ProjectiveHilbertPolynomial,ZZ)
diff(RingElement,Matrix) -- differentiate each entry of a matrix
diff(RingElement,RingElement) -- differentiation
diff(RingElement,Vector) -- differentiate a matrix by a matrix
diff(Vector,Matrix) -- differentiate a matrix by a matrix
diff(Vector,RingElement) -- differentiate a matrix by a matrix
diff(Vector,Vector) -- differentiate a matrix by a matrix
difference -- difference
differentiation
Digamma -- Digamma function
Digamma(RR) -- Digamma function
dim -- compute the Krull dimension
dim(AffineVariety) -- dimension of the affine variety
dim(FractionField) -- compute the Krull dimension
dim(GaloisField) -- compute the Krull dimension
dim(Ideal) -- compute the Krull dimension
dim(InexactField) -- compute the Krull dimension
dim(Module) -- compute the Krull dimension
dim(MonomialIdeal) -- compute the Krull dimension
dim(PolynomialRing) -- compute the Krull dimension
dim(ProjectiveHilbertPolynomial) -- the degree of the Hilbert polynomial
dim(ProjectiveVariety) -- dimension of the projective variety
dim(QuotientRing) -- compute the Krull dimension
dim(Ring) -- compute the Krull dimension
dimension, codimension, and degree
directSum -- direct sum of modules or maps
directSum(ChainComplex) -- direct sum of modules or maps
directSum(GradedModule) -- direct sum of modules or maps
directSum(List) -- direct sum of modules or maps
directSum(Matrix) -- direct sum of modules or maps
directSum(Module) -- direct sum of modules or maps
directSum(Option) -- direct sum of modules or maps
directSum(Sequence) -- direct sum of modules or maps
disassemble -- disassemble pseudocode or a function
dismiss -- dismiss a package
dismiss(Package) -- dismiss a package
dismiss(String) -- dismiss a package
Dispatch -- make a new method function
Divide -- the class of all divide expressions
divideByVariable -- divide all columns by a (power of a) variable
divideByVariable(Matrix,RingElement) -- divide all columns by a (power of a) variable
divideByVariable(Matrix,RingElement,ZZ) -- divide all columns by a (power of a) variable
DivideConquer -- an optional argument
division in polynomial rings with monomials less than 1
do -- loop control
document -- create a documentation node
document(...,Acknowledgement=>...) -- acknowledgements listed in a documentation node
document(...,BaseFunction=>...) -- the basename of a function in a documentation node
document(...,Caveat=>...) -- warnings or limitations listed in a documentation node
document(...,Consequences=>...) -- side-effects of a function in a documentation node
document(...,Contributors=>...) -- non-author contributors listed in a documentation node
document(...,ExampleFiles=>...) -- data files corresponding to a documentation node
document(...,Headline=>...) -- headline of a documentation node
document(...,Inputs=>...) -- inputs for a function in a documentation node
document(...,Key=>...) -- key of a documentation node
document(...,Outputs=>...) -- outputs for a function in a documentation node
document(...,References=>...) -- references listed in a documentation node
document(...,SeeAlso=>...) -- cross-references to other documentation nodes
document(...,SourceCode=>...) -- sources displayed in a documentation node
document(...,Subnodes=>...) -- a menu of documentation nodes
document(...,Usage=>...) -- ways to use a function in a documentation node
document(List) -- create a documentation node
documentation keys
DocumentTag -- the class of all document tags
Down -- monomial ordering
drop -- drop some elements from a list or sequence.
drop(BasicList,List) -- drop some elements from a list or sequence.
drop(BasicList,ZZ) -- drop some elements from a list or sequence.
dual -- dual module or map
dual(BettiTally) -- the class of all Betti tallies
dual(ChainComplex) -- dual
dual(ChainComplexMap) -- dual of a chain complex
dual(CoherentSheaf) -- dual coherent sheaf
dual(Matrix) -- dual of a map
dual(Module) -- dual module
dual(MonomialIdeal) -- the Alexander dual of a monomial ideal
dual(MonomialIdeal,List) -- the Alexander dual
dual(MonomialIdeal,List,Strategy=>...)
dual(MonomialIdeal,RingElement) -- the Alexander dual
dual(MonomialIdeal,RingElement,Strategy=>...)
dual(MonomialIdeal,Strategy=>...)
Dynamic
eagonNorthcott -- Eagon-Northcott complex of a matrix of linear forms
eagonNorthcott(Matrix) -- Eagon-Northcott complex of a matrix of linear forms
echoOff -- turn off echoing
echoOff(File) -- turn off echoing
echoOn -- turn on echoing
echoOn(File) -- turn on echoing
edit -- edit source code
editing Macaulay2 code with emacs
eigenvalues -- find eigenvalues of a matrix
eigenvalues(...,Hermitian=>...) -- Hermitian=>true means assume the matrix is symmetric or Hermitian
eigenvalues(Matrix) -- find eigenvalues of a matrix
eigenvalues(MutableMatrix) -- find eigenvalues of a matrix
eigenvectors -- find eigenvectors of a matrix over RR or CC
eigenvectors(...,Hermitian=>...) -- Hermitian=>true means assume the matrix is symmetric or Hermitian
eigenvectors(Matrix) -- find eigenvectors of a matrix over RR or CC
eigenvectors(MutableMatrix) -- find eigenvectors of a matrix over RR or CC
eint -- exponential integral
eint(RR) -- exponential integral
elapsedTime -- time a computation using time elapsed
elapsedTiming -- time a computation using time elapsed
Elementary uses of Groebner bases I. Math 634 Fall 2005
elements -- list of elements
elements(Set) -- list of elements
elements(Tally) -- list of elements
Eliminate -- elimination order
elimination of variables
else -- condition testing
Email -- the preamble of a package
End -- module of endomorphisms
end -- stop loading a file
endl -- end an output line
endPackage -- end a new package
endPackage(String) -- end a new package
Engine -- specify whether a ring is handled by the engine
engine
engineDebugLevel -- current engine debugging level
EngineRing -- the class of rings handled by the engine
entries -- lists the entries of a matrix
entries(Matrix) -- lists the entries of a matrix
entries(MutableMatrix) -- lists the entries of a matrix
entries(Vector) -- lists the entries of a vector
environment -- the environment variables
equality and containment
equality and containment of modules
Equation -- the class of all equation expressions
Equation == Equation -- equality
Equation == Expression -- equality
Equation == Holder -- equality
erase -- remove a global symbol
erf -- error function
erfc -- complementary error function
error -- deliver error message
error handling -- signalling and trapping errors
errorDepth -- set the error printing depth
euler -- Euler characteristic
euler(CoherentSheaf) -- Euler characteristic of coherent sheaf
euler(Ideal) -- Euler characteristic
euler(Module) -- Euler characteristic
euler(ProjectiveHilbertPolynomial) -- constant term of the Hilbert polynomial
euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
euler(Ring) -- Euler characteristic
EulerConstant -- the Euler-Mascheroni constant
eulers -- list the sectional Euler characteristics
eulers(CoherentSheaf) -- list the sectional Euler characteristics
eulers(Ideal) -- list the sectional Euler characteristics
eulers(Module) -- list the sectional Euler characteristics
eulers(Ring) -- list the sectional Euler characteristics
evaluation and composition of ring maps
even -- tell whether an integer is even
EXAMPLE -- hypertext EXAMPLE item
EXAMPLE(PRE) -- hypertext EXAMPLE item
EXAMPLE(String) -- hypertext EXAMPLE item
EXAMPLE(VisibleList) -- hypertext EXAMPLE item
ExampleFiles -- data files corresponding to a documentation node
examples -- list the examples in documentation
examples of specifying alternate monomial orders
examples(Thing) -- list the examples in documentation
Exclude -- an optional argument
exec -- execute another program
exit -- exit the program
exp -- exponential function
exp(CC) -- exponential function
exp(RingElement)
exp(RR) -- exponential function
exp(RRi) -- exponential function
expm1 -- exponential minus 1
expm1(RR) -- exponential minus 1
expm1(RRi) -- exponential minus 1
exponents -- the exponents of a polynomial
exponents(MonoidElement) -- the exponents of a polynomial
exponents(RingElement) -- the exponents of a polynomial
export -- export symbols from a package
export(List) -- export symbols from a package
export(String) -- export symbols from a package
exportFrom -- export symbols from a package's private dictionary
exportFrom(Package,List) -- export symbols from a package's private dictionary
exportMutable -- export mutable symbols from a package
exportMutable(List) -- export mutable symbols from a package
exportMutable(String) -- export mutable symbols from a package
Expression -- the class of all expressions
expression -- convert to an expression
Expression != Expression -- the class of all expressions
Expression != Holder -- the class of all expressions
Expression != Thing -- the class of all expressions
Expression % Expression -- the class of all expressions
Expression % Holder -- the class of all expressions
Expression % Thing -- the class of all expressions
Expression & Expression -- the class of all expressions
Expression & Holder -- the class of all expressions
Expression & Thing -- the class of all expressions
Expression * Expression -- the class of all expressions
Expression * Holder -- the class of all expressions
Expression * Thing -- the class of all expressions
Expression ** Expression -- the class of all expressions
Expression ** Holder -- the class of all expressions
Expression ** Thing -- the class of all expressions
Expression + Expression -- the class of all expressions
Expression + Holder -- the class of all expressions
Expression + Thing -- the class of all expressions
Expression ++ Expression -- the class of all expressions
Expression ++ Holder -- the class of all expressions
Expression ++ Thing -- the class of all expressions
Expression - Expression -- the class of all expressions
Expression - Holder -- the class of all expressions
Expression - Thing -- the class of all expressions
Expression .. Expression -- the class of all expressions
Expression .. Holder -- the class of all expressions
Expression .. Thing -- the class of all expressions
Expression ..< Expression -- the class of all expressions
Expression ..< Holder -- the class of all expressions
Expression ..< Thing -- the class of all expressions
Expression / Expression -- the class of all expressions
Expression / Holder -- the class of all expressions
Expression / Thing -- the class of all expressions
Expression : Expression -- the class of all expressions
Expression : Holder -- the class of all expressions
Expression : Thing -- the class of all expressions
Expression <== Expression -- the class of all expressions
Expression <== Holder -- the class of all expressions
Expression <== Thing -- the class of all expressions
Expression <=== Expression -- the class of all expressions
Expression <=== Holder -- the class of all expressions
Expression <=== Thing -- the class of all expressions
Expression <==> Expression -- the class of all expressions
Expression <==> Holder -- the class of all expressions
Expression <==> Thing -- the class of all expressions
Expression == Equation -- equality
Expression == Expression -- the class of all expressions
Expression == Holder -- the class of all expressions
Expression == Thing -- the class of all expressions
Expression ===> Expression -- the class of all expressions
Expression ===> Holder -- the class of all expressions
Expression ===> Thing -- the class of all expressions
Expression ==> Expression -- the class of all expressions
Expression ==> Holder -- the class of all expressions
Expression ==> Thing -- the class of all expressions
Expression @ Expression -- the class of all expressions
Expression @ Holder -- the class of all expressions
Expression @ Thing -- the class of all expressions
Expression @@ Expression -- the class of all expressions
Expression @@ Holder -- the class of all expressions
Expression @@ Thing -- the class of all expressions
Expression ^ Expression -- the class of all expressions
Expression ^ Holder -- the class of all expressions
Expression ^ Thing -- the class of all expressions
Expression ^** Expression -- the class of all expressions
Expression ^** Holder -- the class of all expressions
Expression ^** Thing -- the class of all expressions
Expression ^^ Expression -- the class of all expressions
Expression ^^ Holder -- the class of all expressions
Expression ^^ Thing -- the class of all expressions
Expression _ Expression -- the class of all expressions
Expression _ Holder -- the class of all expressions
Expression _ Thing -- the class of all expressions
Expression | Expression -- the class of all expressions
Expression | Holder -- the class of all expressions
Expression | Thing -- the class of all expressions
Expression |- Expression -- the class of all expressions
Expression |- Holder -- the class of all expressions
Expression |- Thing -- the class of all expressions
Expression || Expression -- the class of all expressions
Expression || Holder -- the class of all expressions
Expression || Thing -- the class of all expressions
Expression and Expression -- the class of all expressions
Expression and Holder -- the class of all expressions
Expression and Thing -- the class of all expressions
Expression Expression -- the class of all expressions
Expression Holder -- the class of all expressions
Expression or Expression -- the class of all expressions
Expression or Holder -- the class of all expressions
Expression or Thing -- the class of all expressions
Expression Thing -- the class of all expressions
Expression xor Expression -- the class of all expressions
Expression xor Holder -- the class of all expressions
Expression xor Thing -- the class of all expressions
Ext -- compute an Ext module
Ext(Ideal,Ideal) -- total Ext module
Ext(Ideal,Module) -- total Ext module
Ext(Ideal,Ring) -- total Ext module
Ext(Module,Ideal) -- total Ext module
Ext(Module,Module) -- total Ext module
Ext(Module,Ring) -- total Ext module
Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext
Ext^ZZ(CoherentSheaf,SheafOfRings) -- global Ext
Ext^ZZ(CoherentSheaf,SumOfTwists) -- global Ext
Ext^ZZ(Ideal,Ideal) -- Ext module
Ext^ZZ(Ideal,Matrix) -- map between Ext modules
Ext^ZZ(Ideal,Module) -- Ext module
Ext^ZZ(Ideal,Ring) -- Ext module
Ext^ZZ(Matrix,Ideal) -- map between Ext modules
Ext^ZZ(Matrix,Module) -- map between Ext modules
Ext^ZZ(Matrix,Ring) -- map between Ext modules
Ext^ZZ(Module,Ideal) -- Ext module
Ext^ZZ(Module,Matrix) -- map between Ext modules
Ext^ZZ(Module,Module) -- Ext module
Ext^ZZ(Module,Ring) -- Ext module
Ext^ZZ(SheafOfRings,CoherentSheaf) -- global Ext
Ext^ZZ(SheafOfRings,SheafOfRings) -- global Ext
Ext^ZZ(SheafOfRings,SumOfTwists) -- global Ext
extend -- extend a module map to a chain map, if possible
extend(...,Verify=>...) -- extend a module map to a chain map, if possible
extend(ChainComplex,ChainComplex,Matrix) -- extend a module map to a chain map, if possible
exterior algebras
exterior power of a matrix
exterior power of a module
exteriorPower -- exterior power
exteriorPower(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
exteriorPower(ZZ,CoherentSheaf) -- exterior power
exteriorPower(ZZ,Matrix) -- exterior power of a matrix
exteriorPower(ZZ,Module) -- exterior power of a module
extracting elements
extracting generators of an ideal
extracting information about a matrix
extracting information from chain complexes
factor -- factor a ring element or a ZZ-module
factor(Module) -- factor a ZZ-module
factor(QQ) -- factor a ring element
factor(RingElement) -- factor a ring element
factor(ZZ) -- factor a ring element
factoring polynomials
false
Fano -- Fano scheme
Fano(ZZ,Ideal) -- Fano scheme
Fano(ZZ,Ideal,Ring) -- Fano scheme
FastNonminimal -- compute a non-minimal graded free resolution
fflas-ffpack
File -- the class of all files
File << Manipulator -- print to a file
File << Net -- print to a file
File << String -- print to a file
File << Symbol -- print to a file
File << Thing -- print to a file
file manipulation -- Unix file manipulation functions
fileDictionaries -- local dictionaries for loaded files
fileExecutable
fileExecutable(String)
fileExists -- whether a file exists
fileExitHooks -- a list of hooks (functions) to execute when the current file has been loaded
fileLength -- the length of a file
fileMode -- set or get file mode
fileMode(File) -- get file mode
fileMode(String) -- get file mode
fileMode(ZZ,File) -- set file mode
fileMode(ZZ,String) -- set file mode
FileName -- an optional argument
FilePosition -- the class of all file positions
fileReadable
fileReadable(String)
fileTime -- get or set file modified time
fileWritable
fileWritable(String)
fillMatrix -- fill a mutable matrix with random numbers
fillMatrix(...,Density=>...) -- fill a mutable matrix with random numbers
fillMatrix(...,Height=>...) -- fill a mutable matrix with random numbers
fillMatrix(...,UpperTriangular=>...) -- fill a mutable matrix with random numbers
fillMatrix(MutableMatrix) -- fill a mutable matrix with random numbers
fillMatrix(MutableMatrix,ZZ) -- fill a mutable matrix with random numbers
findFiles -- find files recursively
findFiles(...,Exclude=>...) -- find files recursively
findFiles(...,FollowLinks=>...) -- find files recursively
findFiles(List) -- find files recursively
findFiles(String) -- find files recursively
findHeft -- find a heft vector for a list of degrees
findHeft(...,DegreeRank=>...) -- find a heft vector for a list of degrees
findHeft(List) -- find a heft vector for a list of degrees
finding the Macaulay2 files
finding your files under cygwin
FindOne -- find a primitive element
findProgram -- load external program
findProgram(...,AdditionalPaths=>...) -- load external program
findProgram(...,MinimumVersion=>...) -- load external program
findProgram(...,Prefix=>...) -- load external program
findProgram(...,RaiseError=>...) -- load external program
findProgram(...,Verbose=>...) -- load external program
findProgram(String,List) -- load external program
findProgram(String,String) -- load external program
findSynonyms -- find synonyms of symbols
findSynonyms(Symbol) -- find synonyms of symbols
fine control of a Groebner basis computation
finite field extensions
finite fields
First -- an optional argument
first -- first element of a list
firstkey -- get the first key
fittingIdeal -- Fitting ideal of a module
fittingIdeal(ZZ,Module) -- Fitting ideal of a module
flagLookup -- flag a symbol
FlatMonoid
flatten -- flatten a nested list or a matrix
flatten(Matrix) -- flatten a nested list or a matrix
flatten(VisibleList) -- flatten a nested list or a matrix
flattenRing -- write a ring as a (quotient of a) polynomial ring
flattenRing(...,CoefficientRing=>...) -- optionally specify the desired coefficient ring of the flattened ring
flattenRing(...,Result=>...) -- optionally specify which output(s) to return (see Description for details)
flattenRing(GaloisField) -- write a ring as a (quotient of a) polynomial ring
flattenRing(Ideal) -- write a ring as a (quotient of a) polynomial ring
flattenRing(PolynomialRing) -- write a ring as a (quotient of a) polynomial ring
flattenRing(QuotientRing) -- write a ring as a (quotient of a) polynomial ring
flattenRing(Ring) -- write a ring as a (quotient of a) polynomial ring
Flexible -- a hashtable with information about Macaulay2 operators
FLINT
flip -- matrix of commutativity of tensor product
flip(Module,Module) -- matrix of commutativity of tensor product
floor -- floor function
floor(Constant) -- floor function
floor(Number) -- floor function
flush -- flush output to file
fold -- apply a binary operator repeatedly
fold(Function,Thing) -- apply a binary operator repeatedly
fold(Function,Thing,Thing) -- apply a binary operator repeatedly
fold(Function,VisibleList) -- apply a binary operator repeatedly
fold(VisibleList,Function) -- apply a binary operator repeatedly
fold(VisibleList,Thing,Function) -- apply a binary operator repeatedly
FollowLinks -- an optional argument
for -- for loops
forceGB -- declare that the columns of a matrix are a Gröbner basis
forceGB(...,ChangeMatrix=>...) -- inform Macaulay2 about the change of basis matrix from GB to generators
forceGB(...,MinimalMatrix=>...) -- specify the minimal generator matrix
forceGB(...,SyzygyMatrix=>...) -- specify the syzygy matrix
forceGB(Matrix) -- declare that the columns of a matrix are a Gröbner basis
fork -- fork the process
format -- format a string or a real number
format and display of matrices in Macaulay2
format(CC) -- format a string or a real number
format(DocumentTag) -- the class of all document tags
format(RR) -- format a string or a real number
format(String) -- format a string or a real number
formation -- recover the methods used to make a module
formation(ChainComplex) -- recover the methods used to make a module
formation(ChainComplexMap) -- recover the methods used to make a module
formation(GradedModule) -- recover the methods used to make a module
formation(GradedModuleMap) -- recover the methods used to make a module
formation(Module) -- recover the methods used to make a module
frac -- construct a fraction field
frac(EngineRing) -- construct a fraction field
frac(FractionField) -- construct a fraction field
frac(Ring) -- construct a fraction field
fraction
fraction fields
fraction(RingElement,RingElement)
FractionField -- the class of all fraction fields
frames -- get the frames associated to a closure
frames(Function) -- get the frames associated to a closure
frames(Pseudocode) -- get the frames associated to a closure
frames(Sequence) -- get the frames associated to a closure
frames(Symbol) -- get the frames associated to a closure
free modules
free resolutions of modules
frobby
from -- loop control
Function -- the class of all functions
Function @@ Function -- composition of functions
Function \ Ideal -- apply a function to generators of an ideal
Function \ String -- apply a function to elements of a list
Function \ VirtualTally -- apply a function to elements of a list
Function \ VisibleList -- apply a function to elements of a list
Function \\ Thing -- apply a function
Function _ Thing -- attach the first argument to a function of two or more arguments
function documentation template
function name documentation template
Function Thing -- function application
FunctionApplication -- the class of all function application expressions
FunctionBody -- the class of function bodies
functionBody -- get the body of a function
FunctionClosure -- the class of all function closures
GaloisField -- the class of all Galois fields
Gamma -- Gamma function
gb -- compute a Gröbner basis
gb(...,Algorithm=>...) -- compute a Gröbner basis
gb(...,BasisElementLimit=>...) -- compute a Gröbner basis
gb(...,ChangeMatrix=>...) -- compute a Gröbner basis
gb(...,CodimensionLimit=>...) -- compute a Gröbner basis
gb(...,DegreeLimit=>...) -- compute a Gröbner basis
gb(...,GBDegrees=>...) -- compute a Gröbner basis
gb(...,HardDegreeLimit=>...) -- compute a Gröbner basis
gb(...,Hilbert=>...) -- compute a Gröbner basis
gb(...,MaxReductionCount=>...) -- compute a Gröbner basis
gb(...,PairLimit=>...) -- compute a Gröbner basis
gb(...,StopBeforeComputation=>...) -- compute a Gröbner basis
gb(...,StopWithMinimalGenerators=>...) -- compute a Gröbner basis
gb(...,Strategy=>...) -- compute a Gröbner basis
gb(...,SubringLimit=>...) -- compute a Gröbner basis
gb(...,Syzygies=>...) -- compute a Gröbner basis
gb(...,SyzygyLimit=>...) -- compute a Gröbner basis
gb(...,SyzygyRows=>...) -- compute a Gröbner basis
gb(Ideal) -- compute a Gröbner basis
gb(Matrix) -- compute a Gröbner basis
gb(Module) -- compute a Gröbner basis
GBDegrees -- an optional argument
gbRemove -- remove Gröbner basis
gbRemove(Ideal) -- remove Gröbner basis
gbRemove(Matrix) -- remove Gröbner basis
gbRemove(Module) -- remove Gröbner basis
gbSnapshot -- the Gröbner basis matrix as so far computed
gbSnapshot(Ideal) -- the Gröbner basis matrix as so far computed
gbSnapshot(Matrix) -- the Gröbner basis matrix as so far computed
gbSnapshot(Module) -- the Gröbner basis matrix as so far computed
gbTrace -- provide tracing output during various computations in the engine.
GC garbage collector
gcd -- greatest common divisor
gcd(List) -- greatest common divisor
gcd(QQ,QQ) -- greatest common divisor
gcd(QQ,ZZ) -- greatest common divisor
gcd(RingElement,RingElement) -- greatest common divisor
gcd(RingElement,ZZ) -- greatest common divisor
gcd(Sequence) -- greatest common divisor
gcd(ZZ,QQ) -- greatest common divisor
gcd(ZZ,RingElement) -- greatest common divisor
gcd(ZZ,ZZ) -- greatest common divisor
gcdCoefficients -- gcd with coefficients
gcdCoefficients(RingElement,RingElement) -- gcd with coefficients
gcdCoefficients(ZZ,ZZ) -- gcd with coefficients
GCstats -- information about the status of the garbage collector
genera -- list of the successive linear sectional arithmetic genera
genera(CoherentSheaf) -- list of the successive linear sectional arithmetic genera
genera(Ideal) -- list of the successive linear sectional arithmetic genera
genera(Module) -- list of the successive linear sectional arithmetic genera
genera(ProjectiveVariety) -- list of the successive linear sectional arithmetic genera
genera(Ring) -- list of the successive linear sectional arithmetic genera
GeneralOrderedMonoid -- the class of all ordered free commutative monoids
GenerateAssertions -- trace a function each time it's run
generateAssertions -- generate assert statements from experimental input
generateAssertions(String) -- generate assert statements from experimental input
generator -- provide a single generator
generator(Ideal) -- provide a single generator
generator(Module) -- provide a single generator
generators -- provide matrix or list of generators
generators of ideals and modules
generators(...,CoefficientRing=>...) -- the list of generators of a ring
generators(GroebnerBasis) -- the generator matrix of a Gröbner basis
generators(Ideal) -- the generator matrix of an ideal
generators(Module) -- the generator matrix of a module
generators(Monoid) -- list of generators
generators(MonomialIdeal) -- the generator matrix of an ideal
generators(Ring) -- the list of generators of a ring
Generic -- an optional argument
genericMatrix -- make a generic matrix of variables
genericMatrix(Ring,RingElement,ZZ,ZZ) -- make a generic matrix of variables
genericMatrix(Ring,ZZ,ZZ) -- make a generic matrix of variables
genericSkewMatrix -- make a generic skew symmetric matrix of variables
genericSkewMatrix(Ring,RingElement,ZZ) -- make a generic skew symmetric matrix of variables
genericSkewMatrix(Ring,ZZ) -- make a generic skew symmetric matrix of variables
genericSymmetricMatrix -- make a generic symmetric matrix
genericSymmetricMatrix(Ring,RingElement,ZZ) -- make a generic symmetric matrix
genericSymmetricMatrix(Ring,ZZ) -- make a generic symmetric matrix
genus -- arithmetic genus
genus(CoherentSheaf) -- arithmetic genus
genus(Ideal) -- arithmetic genus
genus(Module) -- arithmetic genus
genus(ProjectiveVariety) -- arithmetic genus
genus(Ring) -- arithmetic genus
get -- get the contents of a file
get(File) -- get the contents of a file
get(String) -- get the contents of a file
getc -- get a byte
getc(File) -- get a byte
getChangeMatrix -- get the change of basis matrix
getChangeMatrix(GroebnerBasis) -- get the change of basis matrix
getenv -- get value of environment variable
getenv(String) -- get value of environment variable
getGlobalSymbol -- create a global symbol in a global dictionary
getNetFile -- get the sequence of completed lines (nets) from a net file
getNonUnit -- retrieve a previously discovered non-unit
getPrimeWithRootOfUnity -- find a prime p with a primitive n-th root of unity r in ZZ/p
getPrimeWithRootOfUnity(...,Range=>...) -- find a prime p with a primitive n-th root of unity r in ZZ/p
getPrimeWithRootOfUnity(ZZ,ZZ) -- find a prime p with a primitive n-th root of unity r in ZZ/p
getSymbol -- make a global user symbol from a string
getting help or reporting bugs
getting input from the user
getting started
getWWW -- get a web page
getWWW(String) -- get a web page
getWWW(String,Nothing) -- get a web page
getWWW(String,String) -- get a web page
GF -- make a finite field
GF(...,PrimitiveElement=>...) -- make a finite field
GF(...,SizeLimit=>...) -- make a finite field
GF(...,Strategy=>...) -- make a finite field
GF(...,Variable=>...) -- make a finite field
GF(Ring) -- make a finite field
GF(ZZ) -- make a finite field
GF(ZZ,ZZ) -- make a finite field
gfan
givaro
GLex -- graded lexicographic ordering
Global -- monomial ordering
global -- get a global symbol
globalAssign -- global assignment, in function form
globalAssignFunction -- the standard method for the global assignment hook
GlobalAssignHook -- hook for assignment to global variables
globalAssignment -- install standard global assignment method
globalAssignmentHooks -- assignment hooks for global symbols
GlobalDictionary -- the class of all global dictionaries
GlobalHookStore -- add a hook function to an object for later processing
globalReleaseFunction -- the standard method for the global variable release hook
GlobalReleaseHook
GNU MP
Gröbner bases
graded and multigraded polynomial rings
graded modules
GradedModule -- the class of all graded modules
gradedModule -- make a graded module
GradedModule ** ChainComplex -- tensor product
GradedModule ** GradedModule -- a binary operator, usually used for tensor product or Cartesian product
GradedModule ** Module -- a binary operator, usually used for tensor product or Cartesian product
GradedModule ++ GradedModule -- direct sum
GradedModule ++ Module -- direct sum of modules
GradedModule == GradedModule -- equality
GradedModule _ ZZ -- component
GradedModule Array -- degree shift
gradedModule(ChainComplex) -- make a graded module
gradedModule(List) -- make a graded module
gradedModule(Module) -- make a graded module
gradedModule(Sequence) -- make a graded module
GradedModuleMap -- the class of all maps between graded modules
gradedModuleMap -- make a map of graded modules
GradedModuleMap * GradedModuleMap -- a binary operator, usually used for multiplication
GradedModuleMap + GradedModuleMap -- a unary or binary operator, usually used for addition
GradedModuleMap + RingElement -- a unary or binary operator, usually used for addition
GradedModuleMap ++ GradedModuleMap -- direct sum of maps
GradedModuleMap - GradedModuleMap -- a unary or binary operator, usually used for negation or subtraction
GradedModuleMap - RingElement -- a unary or binary operator, usually used for negation or subtraction
GradedModuleMap == GradedModuleMap -- equality
GradedModuleMap == RingElement -- equality
GradedModuleMap == ZZ -- equality
GradedModuleMap ^ Array -- component of map corresponding to summand of target
GradedModuleMap ^ ZZ -- iterated composition
GradedModuleMap _ Array -- component of map corresponding to summand of source
GradedModuleMap _ ZZ -- component map
GradedModuleMap | GradedModuleMap -- a binary operator, often used for horizontal concatenation
GradedModuleMap || GradedModuleMap -- a binary operator, often used for vertical concatenation
gradedModuleMap(List) -- make a map of graded modules
gradedModuleMap(Matrix) -- make a map of graded modules
gradedModuleMap(Sequence) -- make a map of graded modules
graphIdeal -- the ideal of the graph of the regular map corresponding to a ring map
graphIdeal(...,MonomialOrder=>...) -- the ideal of the graph of the regular map corresponding to a ring map
graphIdeal(...,MonomialSize=>...) -- the ideal of the graph of the regular map corresponding to a ring map
graphIdeal(...,VariableBaseName=>...) -- the ideal of the graph of the regular map corresponding to a ring map
graphIdeal(RingMap) -- the ideal of the graph of the regular map corresponding to a ring map
graphRing -- the coordinate ring of the graph of the regular map corresponding to a ring map
graphRing(...,MonomialOrder=>...) -- the coordinate ring of the graph of the regular map corresponding to a ring map
graphRing(...,MonomialSize=>...) -- the coordinate ring of the graph of the regular map corresponding to a ring map
graphRing(...,VariableBaseName=>...) -- the coordinate ring of the graph of the regular map corresponding to a ring map
graphRing(RingMap) -- the coordinate ring of the graph of the regular map corresponding to a ring map
Grassmannian -- the Grassmannian of linear subspaces of a vector space
Grassmannian(...,CoefficientRing=>...) -- the Grassmannian of linear subspaces of a vector space
Grassmannian(...,Variable=>...) -- the Grassmannian of linear subspaces of a vector space
Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space
Grassmannian(ZZ,ZZ,PolynomialRing) -- the Grassmannian of linear subspaces of a vector space
GRevLex -- graded reverse lexicographical monomial order.
Groebner basis examples and applications
GroebnerBasis -- the class of all Gröbner bases
groebnerBasis -- Gröbner basis, as a matrix
groebnerBasis(...,Strategy=>...) -- Gröbner basis, as a matrix
groebnerBasis(Ideal) -- Gröbner basis, as a matrix
groebnerBasis(Matrix) -- Gröbner basis, as a matrix
groebnerBasis(Module) -- Gröbner basis, as a matrix
GroebnerBasisOptions
groupID -- the process group identifier
GroupLex -- defines a ring where some variables are inverted
GroupRevLex
handleInterrupts
handling hypertext
HardDegreeLimit -- compute only up to this degree
hash -- hash code of an object
hash tables
hashing
HashTable -- the class of all hash tables
hashTable -- make a hash table
HashTable # Thing -- get value from list, hash table, database, or string
HashTable #? Thing -- check existence of value in a list, hash table, database, or string
hashTable(List) -- make a hash table
HeaderType -- a class of lists with abbreviated constructors
HeaderType List -- a class of lists with abbreviated constructors
HeaderType Sequence -- a class of lists with abbreviated constructors
Heading -- a standardized synopsis for use in documentation nodes
Headline -- headline of a documentation node
Heft -- an optional argument
heft -- heft vector of ring or monoid
heft vectors
heft(Monoid) -- heft vector of ring or monoid
heft(PolynomialRing) -- heft vector of ring or monoid
heft(QuotientRing) -- heft vector of ring or monoid
heft(Ring) -- heft vector of ring or monoid
Height -- an optional argument
height -- height of a net
height(File) -- get window height
height(Net) -- height of a net
height(String) -- height of a net
help -- view documentation nodes
help(Array) -- view documentation nodes
help(DocumentTag) -- view documentation nodes
help(List) -- view documentation nodes
help(Sequence) -- view documentation nodes
help(String) -- view documentation nodes
help(Symbol) -- view documentation nodes
help(Thing) -- view documentation nodes
help(ZZ) -- search the documentation
Hermitian -- an optional argument
HH -- general homology and cohomology functor
hh -- Hodge numbers of a smooth projective variety
HH ChainComplex -- homology of a chain complex
HH ChainComplexMap -- homology of a chain complex map
hh(Sequence,ProjectiveVariety) -- Hodge numbers of a smooth projective variety
HH^ZZ ChainComplex -- cohomology of a chain complex
HH^ZZ ChainComplexMap -- cohomology of a chain complex map
HH^ZZ CoherentSheaf -- cohomology of a coherent sheaf on a projective variety
HH^ZZ Module -- local cohomology of a module
HH^ZZ SheafOfRings -- cohomology of a sheaf of rings on a projective variety
HH^ZZ SumOfTwists -- coherent sheaf cohomology module
HH^ZZ(ProjectiveVariety,CoherentSheaf) -- cohomology of a coherent sheaf on a projective variety
HH_ZZ ChainComplex -- homology of a chain complex
HH_ZZ ChainComplexMap -- homology of a chain complex map
Hilbert -- an optional argument
Hilbert functions and free resolutions -- including degree and betti numbers
hilbertFunction -- the Hilbert function
hilbertFunction(List,CoherentSheaf) -- the Hilbert function
hilbertFunction(List,Ideal) -- the Hilbert function
hilbertFunction(List,Module) -- the Hilbert function
hilbertFunction(List,ProjectiveVariety) -- the Hilbert function
hilbertFunction(List,Ring) -- the Hilbert function
hilbertFunction(ZZ,CoherentSheaf) -- the Hilbert function
hilbertFunction(ZZ,Ideal) -- the Hilbert function
hilbertFunction(ZZ,Module) -- the Hilbert function
hilbertFunction(ZZ,ProjectiveVariety) -- the Hilbert function
hilbertFunction(ZZ,Ring) -- the Hilbert function
hilbertPolynomial -- compute the Hilbert polynomial
hilbertPolynomial(...,Projective=>...) -- choose how to display the Hilbert polynomial
hilbertPolynomial(CoherentSheaf) -- compute the Hilbert polynomial of the coherent sheaf
hilbertPolynomial(Ideal) -- compute the Hilbert polynomial of the quotient of the ambient ring by the ideal
hilbertPolynomial(Module) -- compute the Hilbert polynomial of the module
hilbertPolynomial(ProjectiveVariety) -- compute the Hilbert polynomial of the projective variety
hilbertPolynomial(Ring) -- compute the Hilbert polynomial of the ring
hilbertPolynomial(ZZ,BettiTally) -- the class of all Betti tallies
hilbertSeries -- compute the Hilbert series
hilbertSeries(...,Order=>...) -- display the truncated power series expansion
hilbertSeries(...,Reduce=>...) -- reduce the Hilbert series
hilbertSeries(Ideal) -- compute the Hilbert series of the quotient of the ambient ring by the ideal
hilbertSeries(Module) -- compute the Hilbert series of the module
hilbertSeries(PolynomialRing) -- compute the Hilbert series of a ring
hilbertSeries(ProjectiveHilbertPolynomial) -- compute the Hilbert series of a projective Hilbert polynomial
hilbertSeries(ProjectiveVariety) -- compute the Hilbert series of a projective variety
hilbertSeries(QuotientRing) -- compute the Hilbert series of a ring
hilbertSeries(ZZ,BettiTally) -- the class of all Betti tallies
hold -- hold something in a holder expression
Holder -- the class of all holder expressions
Holder != Expression -- the class of all expressions
Holder != Holder -- the class of all expressions
Holder % Expression -- the class of all expressions
Holder % Holder -- the class of all expressions
Holder & Expression -- the class of all expressions
Holder & Holder -- the class of all expressions
Holder * Expression -- the class of all expressions
Holder * Holder -- the class of all expressions
Holder ** Expression -- the class of all expressions
Holder ** Holder -- the class of all expressions
Holder + Expression -- the class of all expressions
Holder + Holder -- the class of all expressions
Holder ++ Expression -- the class of all expressions
Holder ++ Holder -- the class of all expressions
Holder - Expression -- the class of all expressions
Holder - Holder -- the class of all expressions
Holder .. Expression -- the class of all expressions
Holder .. Holder -- the class of all expressions
Holder ..< Expression -- the class of all expressions
Holder ..< Holder -- the class of all expressions
Holder / Expression -- the class of all expressions
Holder / Holder -- the class of all expressions
Holder / OneExpression
Holder : Expression -- the class of all expressions
Holder : Holder -- the class of all expressions
Holder <== Expression -- the class of all expressions
Holder <== Holder -- the class of all expressions
Holder <=== Expression -- the class of all expressions
Holder <=== Holder -- the class of all expressions
Holder <==> Expression -- the class of all expressions
Holder <==> Holder -- the class of all expressions
Holder == Equation -- equality
Holder == Expression -- the class of all expressions
Holder == Holder -- the class of all expressions
Holder ===> Expression -- the class of all expressions
Holder ===> Holder -- the class of all expressions
Holder ==> Expression -- the class of all expressions
Holder ==> Holder -- the class of all expressions
Holder @ Expression -- the class of all expressions
Holder @ Holder -- the class of all expressions
Holder @@ Expression -- the class of all expressions
Holder @@ Holder -- the class of all expressions
Holder ^ Expression -- the class of all expressions
Holder ^ Holder -- the class of all expressions
Holder ^** Expression -- the class of all expressions
Holder ^** Holder -- the class of all expressions
Holder ^^ Expression -- the class of all expressions
Holder ^^ Holder -- the class of all expressions
Holder _ Expression -- the class of all expressions
Holder _ Holder -- the class of all expressions
Holder | Expression -- the class of all expressions
Holder | Holder -- the class of all expressions
Holder |- Expression -- the class of all expressions
Holder |- Holder -- the class of all expressions
Holder || Expression -- the class of all expressions
Holder || Holder -- the class of all expressions
Holder and Expression -- the class of all expressions
Holder and Holder -- the class of all expressions
Holder Expression -- the class of all expressions
Holder Holder -- the class of all expressions
Holder or Expression -- the class of all expressions
Holder or Holder -- the class of all expressions
Holder xor Expression -- the class of all expressions
Holder xor Holder -- the class of all expressions
Hom -- module of homomorphisms
Hom(ChainComplex,Module)
Hom(ChainComplexMap,Module) -- induced map on Hom
Hom(CoherentSheaf,CoherentSheaf) -- global Hom
Hom(CoherentSheaf,SheafOfRings) -- global Hom
Hom(Ideal,Ideal) -- module of homomorphisms
Hom(Ideal,Module) -- module of homomorphisms
Hom(Ideal,Ring) -- module of homomorphisms
Hom(Matrix,Matrix) -- induced map on Hom
Hom(Matrix,Module) -- induced map on Hom
Hom(Module,ChainComplex)
Hom(Module,ChainComplexMap) -- induced map on Hom
Hom(Module,Ideal) -- module of homomorphisms
Hom(Module,Matrix) -- induced map on Hom
Hom(Module,Module) -- module of homomorphisms
Hom(Module,Ring) -- module of homomorphisms
Hom(Ring,Ideal) -- module of homomorphisms
Hom(Ring,Module) -- module of homomorphisms
Hom(SheafOfRings,CoherentSheaf) -- global Hom
Hom(SheafOfRings,SheafOfRings) -- global Hom
homeDirectory -- the home directory of the user
HomePage -- the preamble of a package
Homogeneous -- a Strategy option value
Homogeneous2 -- compute a Gröbner basis
homogenize -- homogenize with respect to a variable
homogenize(Ideal,RingElement) -- homogenize with respect to a variable
homogenize(Matrix,RingElement) -- homogenize with respect to a variable
homogenize(Matrix,RingElement,List) -- homogenize with respect to a variable
homogenize(Module,RingElement) -- homogenize with respect to a variable
homogenize(Module,RingElement,List) -- homogenize with respect to a variable
homogenize(RingElement,RingElement) -- homogenize with respect to a variable
homogenize(RingElement,RingElement,List) -- homogenize with respect to a variable
homogenize(Vector,RingElement) -- homogenize with respect to a variable
homogenize(Vector,RingElement,List) -- homogenize with respect to a variable
homology -- general homology functor
homology(Matrix,Matrix) -- homology of a pair of maps
homomorphism -- get the homomorphism from element of Hom
homomorphism' -- get the element of Hom from a homomorphism
homomorphism'(Matrix) -- get the element of Hom from a homomorphism
homomorphism(Matrix) -- get the homomorphism from element of Hom
homomorphisms (maps) between modules -- including elements of modules
hooks -- list hooks attached to a key
hooks(...,Strategy=>...) -- list hooks attached to a key
hooks(HashTable) -- list hooks attached to a key
hooks(List) -- list hooks attached to a key
hooks(Sequence) -- list hooks attached to a key
hooks(Symbol) -- list hooks attached to a key
hooks(Thing) -- list hooks attached to a key
hooks(ZZ) -- list hooks attached to a key
horizontalJoin -- join nets or strings horizontally
horizontalJoin(BasicList) -- join nets or strings horizontally
HorizontalSpace -- an optional argument
how Macaulay2 finds its files
html -- convert to HTML format
httpHeaders -- prepend http headers to a string
httpHeaders(String) -- prepend http headers to a string
hypertext -- prepare hypertext for display
hypertext list format
hypertext(List) -- prepare hypertext for display
hypertext(Sequence) -- prepare hypertext for display
id -- identity map
id _ ChainComplex -- identity map
id _ Module -- identity map
id _ Ring -- identity map
Ideal -- the class of all ideals
ideal -- make an ideal
Ideal * CoherentSheaf -- a binary operator, usually used for multiplication
Ideal * Ideal -- product of ideals
Ideal * Module -- a binary operator, usually used for multiplication
Ideal * MonomialIdeal -- product of ideals
Ideal * Ring -- a binary operator, usually used for multiplication
Ideal * Vector -- a binary operator, usually used for multiplication
Ideal ** Ring -- tensor product
Ideal + Ideal -- sum of ideals
Ideal + MonomialIdeal -- sum of ideals
Ideal + RingElement -- a unary or binary operator, usually used for addition
Ideal / Function -- apply a function to generators of an ideal
Ideal / Ideal -- quotient module
Ideal == Ideal -- equality
Ideal == Module -- equality
Ideal == MonomialIdeal -- equality
Ideal == Ring -- equality
Ideal == ZZ -- equality
Ideal ^ Array -- bracket power of an ideal
Ideal ^ ZZ -- power of an ideal
Ideal _ List -- map from free module to some generators
Ideal _ ZZ
Ideal _* -- get the list of generators of an ideal
ideal(List) -- make an ideal
ideal(Matrix) -- make an ideal
ideal(Module) -- converts a module to an ideal
ideal(MonomialIdeal) -- converts a monomial ideal to an ideal
ideal(Number) -- make an ideal
ideal(QuotientRing) -- returns the defining ideal
ideal(Ring) -- returns the defining ideal
ideal(RingElement) -- make an ideal
ideal(Sequence) -- make an ideal
ideal(Variety) -- returns the defining ideal
ideals
ideals to and from matrices
ideals to and from modules
identity -- the identity function
if -- condition testing
IgnoreExampleErrors -- an optional argument
ii -- the square root of -1
image -- image of a map
image(ChainComplexMap) -- image of a map
image(GradedModuleMap) -- image of a map
image(Matrix) -- image of a map
image(RingElement) -- image of a map
imaginaryPart -- imaginary part
imaginaryPart(Number) -- imaginary part
ImmutableType -- the class of immutable types
importFrom -- import symbols to the current private dictionary
importFrom(Package,List) -- import symbols to the current private dictionary
importFrom(String,List) -- import symbols to the current private dictionary
importing and exporting matrices
in -- a keyword used in for-loops
incomparable -- a result indicating incomparability
independentSets -- some size-maximal independent subsets of variables modulo an ideal
independentSets(...,Limit=>...) -- some size-maximal independent subsets of variables modulo an ideal
independentSets(Ideal) -- some size-maximal independent subsets of variables modulo an ideal
independentSets(MonomialIdeal) -- some size-maximal independent subsets of variables modulo an ideal
indeterminate -- an indeterminate number
IndeterminateNumber -- the class of all indeterminate numbers
index -- numeric index of a ring variable
index(RingElement) -- numeric index of a ring variable
indexComponents -- specify keys for components of a direct sum
IndexedVariable -- the class of all indexed variables
IndexedVariable .. IndexedVariable -- sequences of consecutive indexed variables
IndexedVariable ..< IndexedVariable -- sequences of consecutive indexed variables
IndexedVariable <- Thing -- assignment with left side evaluated
IndexedVariable ? IndexedVariable -- comparison operator
IndexedVariable _ Ring -- get a ring variable by name
IndexedVariableTable
IndexedVariableTable _ Thing
IndexedVariableTable _ Thing = Thing
indices -- indices of a polynomial; also components for a direct sum
indices(Matrix) -- indices of variables occurring in a polynomial
indices(RingElement) -- indices of variables occurring in a polynomial
inducedMap -- compute an induced map
inducedMap(...,Degree=>...) -- specify the degree of a map
inducedMap(...,Verify=>...) -- verify that a map is well-defined
inducedMap(ChainComplex,ChainComplex) -- compute the map induced by the identity
inducedMap(Module,Module) -- compute the map induced by the identity
inducedMap(Module,Module,Matrix) -- compute the induced map
inducedMap(Module,Nothing,Matrix) -- compute the induced map
inducedMap(Nothing,Module,Matrix) -- compute the induced map
inducedMap(Nothing,Nothing,Matrix) -- compute the induced map
inducesWellDefinedMap -- whether a map is well defined
inducesWellDefinedMap(Module,Module,Matrix) -- whether a map is well defined
inducesWellDefinedMap(Module,Nothing,Matrix) -- whether a map is well defined
inducesWellDefinedMap(Nothing,Module,Matrix) -- whether a map is well defined
inducesWellDefinedMap(Nothing,Nothing,Matrix) -- whether a map is well defined
InexactField -- the class of inexact fields
InexactFieldFamily -- the class of all families of inexact fields
InexactFieldFamily ^ ZZ -- a binary operator, usually used for powers
InexactFieldFamily Array -- the standard way to make a polynomial ring
InexactFieldFamily List -- make a local polynomial ring
InexactFieldFamily Monoid -- make a polynomial ring
InexactNumber
InexactNumber * Constant -- a binary operator, usually used for multiplication
InexactNumber + Constant -- a unary or binary operator, usually used for addition
InexactNumber - Constant -- a unary or binary operator, usually used for negation or subtraction
InexactNumber / Constant
InexactNumber == Constant
InexactNumber ^ Constant
InexactNumber'
InfiniteNumber -- the class of all infinite numbers
InfiniteNumber + InfiniteNumber -- a unary or binary operator, usually used for addition
InfiniteNumber + Number -- a unary or binary operator, usually used for addition
InfiniteNumber - InfiniteNumber -- a unary or binary operator, usually used for negation or subtraction
InfiniteNumber - Number -- a unary or binary operator, usually used for negation or subtraction
InfiniteNumber // QQ -- a binary operator, usually used for quotient
InfiniteNumber // RR -- a binary operator, usually used for quotient
InfiniteNumber // ZZ -- a binary operator, usually used for quotient
InfiniteNumber == InfiniteNumber -- equality
InfiniteNumber == Number -- equality
infinity -- infinity
info -- convert hypertext to info format
InfoDirSection -- an optional argument
infoHelp -- view documentation in Info format
infoHelp(DocumentTag) -- view documentation in Info format
infoHelp(Thing) -- view documentation in Info format
infoHelp(ZZ) -- search the documentation
information about a map of modules
informing others about your package
inheritance
Inhomogeneous -- a Strategy option value
initial help
initialization file
input -- read Macaulay2 commands and echo
Inputs -- inputs for a function in a documentation node
inputting a matrix
insert -- copy a list, inserting an element
insert(ZZ,Thing,VisibleList) -- copy a list, inserting an element
installAssignmentMethod -- install methods assignment to the value of an operator
installAssignmentMethod(Symbol,HashTable,Function) -- install methods assignment to the value of an operator
installAssignmentMethod(Symbol,HashTable,HashTable,Function) -- install methods assignment to the value of an operator
installAssignmentMethod(Symbol,HashTable,HashTable,Option) -- install methods assignment to the value of an operator
installAssignmentMethod(Symbol,HashTable,Option) -- install methods assignment to the value of an operator
installedPackages
installHilbertFunction -- assemble degrees of a ring, module, or ideal into a polynomial
installing methods
installMethod -- install methods
installPackage -- load and install a package and its documentation
installPackage(...,CacheExampleOutput=>...) -- load and install a package and its documentation
installPackage(...,CheckDocumentation=>...) -- load and install a package and its documentation
installPackage(...,DebuggingMode=>...) -- load and install a package and its documentation
installPackage(...,FileName=>...) -- load and install a package and its documentation
installPackage(...,IgnoreExampleErrors=>...) -- load and install a package and its documentation
installPackage(...,InstallPrefix=>...) -- load and install a package and its documentation
installPackage(...,MakeDocumentation=>...) -- load and install a package and its documentation
installPackage(...,MakeHTML=>...) -- load and install a package and its documentation
installPackage(...,MakeInfo=>...) -- load and install a package and its documentation
installPackage(...,MakeLinks=>...) -- load and install a package and its documentation
installPackage(...,MakePDF=>...) -- load and install a package and its documentation
installPackage(...,RemakeAllDocumentation=>...) -- load and install a package and its documentation
installPackage(...,RerunExamples=>...) -- load and install a package and its documentation
installPackage(...,RunExamples=>...) -- load and install a package and its documentation
installPackage(...,SeparateExec=>...) -- load and install a package and its documentation
installPackage(...,UserMode=>...) -- load and install a package and its documentation
installPackage(...,Verbose=>...) -- load and install a package and its documentation
installPackage(Package) -- load and install a package and its documentation
installPackage(String) -- load and install a package and its documentation
InstallPrefix -- an optional argument
instance -- whether something has a certain type
instance(Thing,Type) -- whether something has a certain type
instances
instances(Type)
integers modulo a prime
integrate -- numerical integration
integrate(Function,Constant,Constant) -- numerical integration
integrate(Function,Constant,InfiniteNumber) -- numerical integration
integrate(Function,Constant,Number) -- numerical integration
integrate(Function,InfiniteNumber,Constant) -- numerical integration
integrate(Function,InfiniteNumber,InfiniteNumber) -- numerical integration
integrate(Function,InfiniteNumber,Number) -- numerical integration
integrate(Function,Number,Constant) -- numerical integration
integrate(Function,Number,InfiniteNumber) -- numerical integration
integrate(Function,Number,Number) -- numerical integration
interpreterDepth -- nesting depth of the interpreter
intersect -- compute an intersection
intersect(Ideal) -- compute an intersection of a sequence of ideals or modules
intersect(Ideal,Ideal) -- compute an intersection of a sequence of ideals or modules
intersect(Ideal,Ideal,MinimalGenerators=>...) -- compute an intersection of a sequence of ideals or modules
intersect(Ideal,Ideal,Strategy=>...) -- compute an intersection of a sequence of ideals or modules
intersect(List) -- compute an intersection
intersect(Module) -- compute an intersection of a sequence of ideals or modules
intersect(Module,Module) -- compute an intersection of a sequence of ideals or modules
intersect(Module,Module,MinimalGenerators=>...) -- compute an intersection of a sequence of ideals or modules
intersect(Module,Module,Strategy=>...) -- compute an intersection of a sequence of ideals or modules
intersect(RRi) -- computes intersection of input intervals
intersect(RRi,Precision=>...) -- computes intersection of input intervals
intersect(Sequence) -- compute an intersection
Intersection -- value of an optional argument
intersection -- compute an intersection
intersection of ideals
intersection(RRi,RRi) -- computes intersection of input intervals
intersection(RRi,RRi,Precision=>...) -- computes intersection of input intervals
interval -- construct an interval
interval(...,Precision=>...) -- construct an interval
interval(Array) -- construct an interval
interval(QQ) -- construct an interval
interval(QQ,QQ) -- construct an interval
interval(QQ,RR) -- construct an interval
interval(QQ,ZZ) -- construct an interval
interval(RR) -- construct an interval
interval(RR,QQ) -- construct an interval
interval(RR,RR) -- construct an interval
interval(RR,ZZ) -- construct an interval
interval(ZZ) -- construct an interval
interval(ZZ,QQ) -- construct an interval
interval(ZZ,RR) -- construct an interval
interval(ZZ,ZZ) -- construct an interval
inverse -- compute the inverse
inverse(Matrix) -- compute the inverse
inverse(MutableMatrix) -- compute the inverse
inverse(RingMap) -- compute the inverse
inverseErf -- inverse error function
InverseMethod -- compute reciprocals
inversePermutation -- inverse permutation
inverseRegularizedBeta -- inverse of the regularized beta function
inverseRegularizedGamma -- inverse of the upper regularized gamma function
Inverses -- an optional argument
Invertible -- an optional argument
Invoking the program
irreducibleCharacteristicSeries -- irreducible characteristic series of an ideal
irreducibleCharacteristicSeries(Ideal) -- irreducible characteristic series of an ideal
isAffineRing -- whether something is an affine ring
isAffineRing(PolynomialRing) -- whether something is an affine ring
isAffineRing(QuotientRing) -- whether something is an affine ring
isAffineRing(Ring) -- whether something is an affine ring
isANumber -- whether a number is not not-a-number
isANumber(Number) -- whether a number is not not-a-number
isBorel -- whether an ideal is fixed by upper triangular changes of coordinates
isBorel(MonomialIdeal) -- whether an ideal is fixed by upper triangular changes of coordinates
isCanceled -- whether a task has been canceled
isCanceled(Task) -- whether a task has been canceled
isCommutative -- whether a ring is commutative
isCommutative(Ring) -- whether a ring is commutative
isConstant -- whether a ring element is constant
isConstant(RingElement) -- whether a ring element is constant
isDirectory -- whether a file is a directory
isDirectSum -- whether something is a direct sum
isDirectSum(ChainComplex) -- whether something is a direct sum
isDirectSum(GradedModule) -- whether something is a direct sum
isDirectSum(Module) -- whether something is a direct sum
isEmpty
isEmpty(RRi) -- empty test for an interval
isField -- whether something is a field
isField(EngineRing) -- whether something is a field
isField(Ring) -- whether something is a field
isFinite -- whether a number is finite
isFinite(Number) -- whether a number is finite
isFinitePrimeField -- whether a ring is a finite prime field
isFreeModule -- whether something is a free module
isFreeModule(Module) -- whether something is a free module
isFreeModule(Thing) -- whether something is a free module
isGlobalSymbol -- whether a global symbol with a given name exists
isHomogeneous -- whether something is homogeneous (graded)
isHomogeneous(ChainComplex) -- whether something is homogeneous (graded)
isHomogeneous(ChainComplexMap) -- whether something is homogeneous (graded)
isHomogeneous(Ideal) -- whether something is homogeneous (graded)
isHomogeneous(Matrix) -- whether something is homogeneous (graded)
isHomogeneous(Module) -- whether something is homogeneous (graded)
isHomogeneous(Ring) -- whether something is homogeneous (graded)
isHomogeneous(RingElement) -- whether something is homogeneous (graded)
isHomogeneous(RingMap) -- whether something is homogeneous (graded)
isHomogeneous(Vector) -- whether something is homogeneous (graded)
isIdeal -- whether something is an ideal
isIdeal(Ideal) -- whether something is an ideal
isIdeal(Module) -- whether something is an ideal
isIdeal(MonomialIdeal) -- whether something is an ideal
isIdeal(Thing) -- whether something is an ideal
isInfinite -- whether a number is infinite
isInfinite(Constant) -- whether a number is infinite
isInfinite(InfiniteNumber) -- whether a number is infinite
isInfinite(Number) -- whether a number is infinite
isInjective -- whether a map is injective
isInjective(Matrix) -- whether a map is injective
isInjective(RingMap) -- whether a map is injective
isInputFile -- whether a file is open for input
isInputFile(File) -- whether a file is open for input
isIsomorphism -- whether a map is an isomorphism
isIsomorphism(Matrix) -- whether a map is an isomorphism
isListener -- whether a file is open for listening
isListener(File) -- whether a file is open for listening
isMember -- membership test in an interval
isMember(QQ,RRi) -- membership test in an interval
isMember(RR,RRi) -- membership test in an interval
isMember(ZZ,RRi) -- membership test in an interval
isModule -- whether something is a module
isModule(Module) -- whether something is a module
isModule(Thing) -- whether something is a module
isMonomialIdeal -- whether something is a monomial ideal
isMonomialIdeal(Ideal) -- whether something is a monomial ideal
isMonomialIdeal(MonomialIdeal) -- whether something is a monomial ideal
isMonomialIdeal(Thing) -- whether something is a monomial ideal
isOpen -- whether a file or database is open
isOpen(Database) -- whether a file or database is open
isOpen(File) -- whether a file or database is open
isOutputFile -- whether a file is open for output
isOutputFile(File) -- whether a file is open for output
isPolynomialRing -- whether something is a polynomial ring
isPolynomialRing(PolynomialRing) -- whether something is a polynomial ring
isPolynomialRing(Thing) -- whether something is a polynomial ring
isPrime -- whether a integer or polynomial is prime
isPrime(ZZ) -- whether a integer or polynomial is prime
isPrimitive -- whether an element is a primitive element of a finite field
isPseudoprime -- whether an integer is probably prime
isPseudoprime(ZZ) -- whether an integer is probably prime
isQuotientModule -- whether something is evidently a quotient of a free module
isQuotientModule(Module) -- whether something is evidently a quotient of a free module
isQuotientModule(Thing) -- whether something is evidently a quotient of a free module
isQuotientOf -- whether one thing is a quotient of another
isQuotientOf(Ring,QuotientRing) -- whether one ring is a quotient of another
isQuotientOf(Ring,Ring) -- whether one ring is a quotient of another
isQuotientOf(Type,QuotientRing) -- whether one ring is a quotient of a ring of a given type
isQuotientOf(Type,Ring) -- whether one ring is a quotient of a ring of a given type
isQuotientRing -- whether something is a quotient ring
isQuotientRing(QuotientRing) -- whether something is a quotient ring
isQuotientRing(Ring) -- whether something is a quotient ring
isReady -- whether a file has data available for reading
isReady(File) -- whether a file has data available for reading
isReady(Task) -- whether a task is finished
isReal -- whether a number is real
isReal(CC) -- whether a number is real
isReal(Constant) -- whether a number is real
isReal(InfiniteNumber) -- whether a number is real
isReal(QQ) -- whether a number is real
isReal(RR) -- whether a number is real
isReal(RRi) -- whether a number is real
isReal(ZZ) -- whether a number is real
isRegularFile -- whether a file is a regular file
isRing -- whether something is a ring
isRing(Ring) -- whether something is a ring
isRing(Thing) -- whether something is a ring
isSkewCommutative -- whether a ring has skew commuting variables
isSkewCommutative(PolynomialRing) -- whether a ring has skew commuting variables
isSkewCommutative(QuotientRing) -- whether a ring has skew commuting variables
isSkewCommutative(Ring) -- whether a ring has skew commuting variables
isSorted -- whether a list is sorted
isSorted(VisibleList) -- whether a list is sorted
isSquareFree -- whether something is square free monomial ideal
isSquareFree(MonomialIdeal) -- whether something is square free monomial ideal
isSubmodule -- whether a module is evidently a submodule of a free module
isSubmodule(Module) -- whether a module is evidently a submodule of a free module
isSubmodule(Thing) -- whether a module is evidently a submodule of a free module
isSubquotient -- check whether a module is a subquotient of another
isSubquotient(Module,Module) -- check whether a module is a subquotient of another
isSubset -- whether one object is a subset of another
isSubset(Ideal,Ideal) -- whether one object is a subset of another
isSubset(Ideal,Module) -- whether one object is a subset of another
isSubset(Module,Ideal) -- whether one object is a subset of another
isSubset(Module,Module) -- whether one object is a subset of another
isSubset(RRi,RRi) -- subset test for intervals
isSubset(Set,Set) -- whether one object is a subset of another
isSubset(Set,VisibleList) -- whether one object is a subset of another
isSubset(VisibleList,Set) -- whether one object is a subset of another
isSubset(VisibleList,VisibleList) -- whether one object is a subset of another
isSurjective -- whether a map is surjective
isSurjective(Matrix) -- whether a map is surjective
isTable -- whether something is a list of lists of equal length
isUnit -- whether a ring element is a unit
isUnit(Number) -- whether a ring element is a unit
isUnit(RingElement) -- whether a ring element is a unit
isUnit(ZZ) -- whether a ring element is a unit
isWellDefined -- whether a map is well defined
isWellDefined(Matrix) -- whether a map is well defined
isWellDefined(RingMap) -- whether a map is well defined
isWeylAlgebra
Iterate -- value of an optional argument
Iterator -- class for iterators
iterator -- get an iterator
iterator(Iterator) -- get an iterator
iterator(String) -- get an iterator
iterator(VisibleList) -- get an iterator
Jacobian
jacobian -- the Jacobian matrix of partial derivatives
jacobian(Ideal) -- the Jacobian matrix of the generators of an ideal
jacobian(Matrix) -- the matrix of partial derivatives of polynomials in a matrix
jacobian(MonomialIdeal) -- the Jacobian matrix of the generators of an ideal
jacobian(Ring) -- the Jacobian matrix of the polynomials defining a quotient ring
Join -- an optional argument
join -- join lists, sequences, and iterable objects
Jupyter -- top level printing method used in the Jupyter kernel
KeepFiles -- whether to keep temporary files
KeepZeroes -- an optional argument
kernel -- kernel of a ringmap, matrix, or chain complex
kernel and coimage of a ring map
kernel(...,SubringLimit=>...)
kernel(ChainComplexMap) -- kernel of a chain complex map
kernel(GradedModuleMap) -- kernel of a chain complex map
kernel(Matrix) -- kernel of a matrix
kernel(RingElement) -- kernel of a matrix
kernel(RingMap) -- kernel of a ringmap
Key -- key of a documentation node
keys -- keys used in a hash table, dictionary, or database
keys(Database) -- keys used in a hash table, dictionary, or database
keys(Dictionary) -- keys used in a hash table, dictionary, or database
keys(HashTable) -- keys used in a hash table, dictionary, or database
Keyword -- the class of all keywords
Keywords -- an optional argument
kill -- kill a process
kill(File) -- kill a process
kill(ZZ) -- kill a process
koszul -- Koszul complex or specific matrix in the Koszul complex
koszul(Matrix) -- the Koszul complex
koszul(ZZ,Matrix) -- a differential in a Koszul complex
LAPACK
last -- last element of a list
lastMatch -- regular expression matching
Layout -- relative locations of Macaulay2 files
lcm -- least common multiple
lcm(List) -- least common multiple
lcm(MonomialIdeal) -- least common multiple of all minimal generators
lcm(QQ,QQ) -- least common multiple
lcm(QQ,ZZ) -- least common multiple
lcm(RingElement,RingElement) -- least common multiple
lcm(RingElement,ZZ) -- least common multiple
lcm(Sequence) -- least common multiple
lcm(ZZ,QQ) -- least common multiple
lcm(ZZ,RingElement) -- least common multiple
lcm(ZZ,ZZ) -- least common multiple
leadCoefficient -- the coefficient of the leading term
leadCoefficient(RingElement) -- the coefficient of the leading term
leadComponent -- the leading component(s) of a vector or matrix
leadComponent(Matrix) -- the leading component(s) of a vector or matrix
leadComponent(Vector) -- the leading component(s) of a vector or matrix
leadMonomial -- the leading monomial of a ring element
leadMonomial(RingElement) -- the leading monomial of a ring element
leadTerm -- get the greatest term
leadTerm(GroebnerBasis) -- get the greatest term of each column
leadTerm(Ideal) -- get the ideal of greatest terms
leadTerm(Matrix) -- get the greatest term of each column
leadTerm(RingElement) -- get the greatest term
leadTerm(Vector) -- get the greatest term of each column
leadTerm(ZZ,Ideal) -- get the ideal of lead polynomials
leadTerm(ZZ,Matrix) -- get the matrix of lead polynomials of each column
leadTerm(ZZ,RingElement) -- get the lead polynomials using part of the monomial order
Left -- value of an optional argument
left -- left endpoint of an interval
left shift
length -- length of an object
length(ChainComplex) -- length of a chain complex or graded module
length(Dictionary) -- length of a dictionary
length(GradedModule) -- length of a graded module
length(Module) -- Computes the length of a module
length(String) -- length of a string
length(VisibleList) -- length of a visible list
LengthLimit -- stop when the resolution reaches this length
Lex -- lexicographical monomial order.
lift -- lift to another ring
lift(...,Verify=>...) -- lift to another ring
lift(BettiTally,type of ZZ) -- the class of all Betti tallies
lift(CC,type of QQ) -- lift to another ring
lift(CC,type of RR_*) -- lift to another ring
lift(CC,type of ZZ) -- lift to another ring
lift(Ideal,type of QQ) -- lift to another ring
lift(Ideal,type of RingElement) -- lift to another ring
lift(Ideal,type of ZZ) -- lift to another ring
lift(Matrix,type of CC_*,type of QQ) -- lift to another ring
lift(Matrix,type of CC_*,type of RR_*) -- lift to another ring
lift(Matrix,type of CC_*,type of ZZ) -- lift to another ring
lift(Matrix,type of Number) -- lift to another ring
lift(Matrix,type of QQ,type of QQ) -- lift to another ring
lift(Matrix,type of QQ,type of ZZ) -- lift to another ring
lift(Matrix,type of RingElement) -- lift to another ring
lift(Matrix,type of RR_*,type of QQ) -- lift to another ring
lift(Matrix,type of RR_*,type of ZZ) -- lift to another ring
lift(Matrix,type of RRi_*,type of QQ) -- lift to another ring
lift(Matrix,type of RRi_*,type of RR_*) -- lift to another ring
lift(Matrix,type of RRi_*,type of ZZ) -- lift to another ring
lift(Matrix,type of ZZ,type of ZZ) -- lift to another ring
lift(QQ,type of QQ) -- lift to another ring
lift(QQ,type of ZZ) -- lift to another ring
lift(RR,type of QQ) -- lift to another ring
lift(RR,type of ZZ) -- lift to another ring
lift(RRi,type of QQ) -- lift to another ring
lift(RRi,type of RR_*) -- lift to another ring
lift(RRi,type of ZZ) -- lift to another ring
lift(ZZ,type of ZZ) -- lift to another ring
liftable -- whether lifting to another ring is possible
Limit -- an optional argument
limitFiles
limitProcesses
Linear -- value of an optional argument
LinearAlgebra -- compute a Gröbner basis
lineNumber -- current line number
lines -- split a string into lines
lines(String) -- split a string into lines
lines(String,String) -- split a string into lines
linkFile -- make a new link to a file
linkFile(String,String) -- make a new link to a file
List -- the class of all lists -- {...}
list -- loop control
List ** List -- Cartesian product of two lists
List + List -- sum of two vectors
List - List -- difference of two vectors
List - Set -- set difference
List .. List -- rectangular sequences of consecutive lists and sequences
List ..< List -- rectangular sequences of consecutive lists and sequences
List / Command -- apply a function to elements of a list
List / Function -- apply a function to elements of a list
List / RingMap -- apply a function to elements of a list
List / Thing -- vector division
List << List -- component-wise comparison of lists
List >> Function -- attaching options to a function
List ? List -- comparison operator
List | List -- join lists, sequences or arrays
list of obsolete functions
listForm -- convert to list form
listForm(MonoidElement) -- convert to list form
listForm(RingElement) -- convert to list form
listLocalSymbols -- display of local symbols and their values
lists and sequences -- a detailed overview of lists and sequences in Macaulay2
listSymbols -- compact display of symbols and their values
listSymbols(Dictionary) -- compact display of symbols and their values
listSymbols(List) -- compact display of symbols and their values
listUserSymbols -- display the user's symbols
lngamma -- logarithm of the Gamma function
lngamma(Constant) -- logarithm of the Gamma function
lngamma(QQ) -- logarithm of the Gamma function
lngamma(RR) -- logarithm of the Gamma function
lngamma(ZZ) -- logarithm of the Gamma function
load -- read Macaulay2 commands
loadDepth -- the load depth
LoadDocumentation -- an optional argument
loadedFiles
loadedPackages -- the list of loaded packages
loadPackage -- load a package
loadPackage(...,Configuration=>...) -- load a package
loadPackage(...,DebuggingMode=>...) -- load a package
loadPackage(...,FileName=>...) -- load a package
loadPackage(...,LoadDocumentation=>...) -- load a package
loadPackage(...,Reload=>...) -- load a package
loadPackage(Package) -- load a package
loadPackage(String) -- load a package
Local -- an optional argument
local -- get a local symbol
local variables in a file
local variables in a function
localDictionaries -- get local dictionaries
localDictionaries(Dictionary) -- get local dictionaries
localDictionaries(Function) -- get local dictionaries
localDictionaries(Pseudocode) -- get local dictionaries
localDictionaries(Symbol) -- get local dictionaries
LocalDictionary -- the class of all local dictionaries
locate -- locate source code
locate(DocumentTag) -- the class of all document tags
locate(Function) -- locate source code
locate(List) -- locate source code
locate(Nothing) -- locate source code
locate(Pseudocode) -- locate source code
locate(Sequence) -- locate source code
locate(Symbol) -- locate source code
locate(TestInput) -- locate a package's tests
locate(ZZ) -- locate source code
log -- logarithm function
log(CC) -- logarithm function
log(RR) -- logarithm function
log(RR,RR) -- logarithm function
log(RRi) -- logarithm function
log(RRi,RRi) -- logarithm function
log1p -- logarithm of 1+x
log1p(RR) -- logarithm of 1+x
log1p(RRi) -- logarithm of 1+x
LongPolynomial -- a Strategy option value
lookup -- look up methods
lookupCount -- reference count for a symbol
LowerBound -- the class of lower bound objects
lrslib
LUdecomposition -- LU decomposition
LUdecomposition(Matrix) -- LU decomposition
LUdecomposition(MutableMatrix) -- LU decomposition
M2SingularBook -- Macaulay2 examples for the Singular book
Macaulay2 -- Macaulay2 documentation
Macaulay2Doc -- Macaulay2 documentation
makeDirectory -- make a directory
makeDirectory(String) -- make a directory
MakeDocumentation -- an optional argument
makeDocumentTag -- the class of all document tags
makeDocumentTag(DocumentTag) -- the class of all document tags
makeDocumentTag(List) -- the class of all document tags
makeDocumentTag(String) -- the class of all document tags
makeDocumentTag(Thing) -- the class of all document tags
MakeHTML -- an optional argument
MakeInfo -- an optional argument
MakeLinks -- an optional argument
makePackageIndex -- create an index of installed packages
makePackageIndex(List) -- create an index of installed packages
makePackageIndex(Sequence) -- create an index of installed packages
MakePDF -- an optional argument
making a new method function
making chain complexes by hand
making functions
making functions with a variable number of arguments
making functions with multiple return values
making modules from matrices
making new classes
making new functions with optional arguments
manipulating chain complexes
manipulating modules
manipulating polynomials
Manipulator -- the class of all file manipulators
Manipulator Database -- the class of all file manipulators
Manipulator File -- the class of all file manipulators
Manipulator NetFile -- the class of all net files
Manipulator Nothing -- the class of all file manipulators
map -- make a map
map(...,Degree=>...) -- specify the degree of a map
map(...,DegreeLift=>...) -- make a ring map
map(...,DegreeMap=>...) -- make a ring map
map(ChainComplex,ChainComplex,Function) -- make a map of chain complexes
map(GradedModule,GradedModule,Function) -- make a map of chain complexes
map(Matrix) -- make a matrix with a different degree
map(Module) -- identity map
map(Module,Module,Function) -- create a matrix by specifying a function that gives each entry
map(Module,Module,List) -- create a matrix by giving a sparse or dense list of entries
map(Module,Module,Matrix) -- create the matrix induced on generators by a given matrix
map(Module,Module,Number) -- construct the map induced by multiplication by a ring element on the generators
map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators
map(Module,Module,RingMap,List) -- homomorphism of modules over different rings
map(Module,Module,RingMap,Matrix) -- homomorphism of modules over different rings
map(Module,Module,ZZ) -- construct the map induced by multiplication by a ring element on the generators
map(Module,Nothing,List) -- create a matrix by giving a doubly nested list of ring elements
map(Module,Nothing,Matrix) -- recast a matrix to have a new target, and a free module as source
map(Module,Nothing,RingMap,List) -- homomorphism of modules over different rings
map(Module,Nothing,RingMap,Matrix) -- homomorphism of modules over different rings
map(Module,RingMap) -- homomorphism of modules over different rings
map(Module,ZZ,Function) -- create a matrix from a free module by specifying a function that gives each entry
map(Module,ZZ,List) -- create a matrix by giving a sparse or dense list of entries
map(Module,ZZ,ZZ) -- construct the map induced by multiplication by a ring element on the generators
map(Ring,Matrix) -- make a ring map
map(Ring,Ring) -- make a ring map, using the names of the variables
map(Ring,Ring,List) -- make a ring map
map(Ring,Ring,Matrix) -- make a ring map
map(Ring,Ring,RingMap) -- make a ring map
MapExpression -- the class of map expressions
mapping over hash tables -- apply a function to each element of a hash table
maps between chain complexes
maps between modules
markedGB -- make a marked Gröbner basis
markedGB(...,ChangeMatrix=>...) -- make a marked Gröbner basis
markedGB(...,MinimalMatrix=>...) -- make a marked Gröbner basis
markedGB(...,SyzygyMatrix=>...) -- make a marked Gröbner basis
markedGB(Matrix,Matrix) -- make a marked Gröbner basis
match -- regular expression matching
match(...,POSIX=>...) -- regular expression matching
match(...,Strategy=>...) -- regular expression matching
match(List,String) -- regular expression matching
match(String,String) -- regular expression matching
mathematical examples
mathML -- convert to MathML format
matrices
matrices to and from modules -- including kernel, cokernel and image
Matrix -- the class of all matrices
matrix -- make a matrix
Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
Matrix % Ideal -- normal form of ring elements and matrices
Matrix % Matrix -- normal form of ring elements and matrices
Matrix % Module -- normal form of ring elements and matrices
Matrix % MonomialIdeal -- normal form of ring elements and matrices
Matrix % RingElement -- normal form of ring elements and matrices
Matrix * Matrix -- matrix multiplication
Matrix * Number -- a binary operator, usually used for multiplication
Matrix * RingElement -- a binary operator, usually used for multiplication
Matrix * Vector -- a binary operator, usually used for multiplication
Matrix ** Matrix -- tensor product
Matrix ** Module -- tensor product
Matrix ** Ring -- tensor product
Matrix ** RingElement -- a binary operator, usually used for tensor product or Cartesian product
Matrix + Matrix -- a unary or binary operator, usually used for addition
Matrix + Number -- a unary or binary operator, usually used for addition
Matrix + RingElement -- a unary or binary operator, usually used for addition
Matrix ++ Matrix -- direct sum of maps
Matrix ++ RingElement -- direct sum of maps
Matrix - Matrix -- a unary or binary operator, usually used for negation or subtraction
Matrix - Number -- a unary or binary operator, usually used for negation or subtraction
Matrix - RingElement -- a unary or binary operator, usually used for negation or subtraction
Matrix // GroebnerBasis -- factor a map through another
Matrix // Matrix -- factor a map through another
Matrix // MonomialIdeal -- factor a map through another
Matrix // Number -- a binary operator, usually used for quotient
Matrix // RingElement -- factor a map through another
Matrix == Matrix -- equality
Matrix == Number -- equality
Matrix == RingElement -- equality
Matrix == ZZ -- equality
Matrix \\ Matrix -- factor a map through another
Matrix \\ RingElement -- factor a map through another
Matrix ^ Array -- component of map corresponding to summand of target
Matrix ^ List -- select rows
Matrix ^ ZZ -- power
Matrix _ Array -- component of map corresponding to summand of source
Matrix _ List -- select columns
Matrix _ Sequence -- get entry of matrix
Matrix _ ZZ
Matrix | Matrix -- join matrices horizontally
Matrix | RingElement -- join matrices horizontally
Matrix | ZZ -- join matrices horizontally
Matrix || Matrix -- join matrices vertically
Matrix || RingElement -- join matrices vertically
Matrix || ZZ -- join matrices vertically
matrix(...,Degree=>...) -- create a matrix from a doubly-nested list of ring elements or matrices
matrix(List) -- create a matrix from a doubly-nested list of ring elements or matrices
matrix(Matrix) -- the matrix between generators
matrix(MutableMatrix) -- make a matrix from a mutable one
matrix(Number) -- make a matrix from a ring element
matrix(Ring,List) -- create a matrix from a doubly nested list of ring elements or matrices
matrix(RingElement) -- make a matrix from a ring element
matrix(RingFamily,List) -- create a matrix from a doubly nested list of ring elements or matrices
matrix(RingMap) -- the matrix associated to a ring map
matrix(Vector) -- make a matrix
MatrixExpression -- the class of all matrix expressions
max -- yields the maximum element in a list or sequence
max(GradedModule)
max(VisibleList) -- yields the maximum element in a list or sequence
maxAllowableThreads -- the maximum possible number of simultaneously running tasks
maxExponent
MaximalRank -- an optional argument
maxPosition -- position of the largest element
maxPosition(BasicList) -- position of the largest element
MaxReductionCount -- an optional argument
member -- test membership in a list or set
member(Thing,Set) -- test membership in a list or set
member(Thing,VisibleList) -- test membership in a list or set
memoize -- record results of function evaluation for future use
memoize(Function) -- record results of function evaluation for future use
memoize(Function,List) -- record results of function evaluation for future use
memoizeClear -- record results of function evaluation for future use
memoizeValues -- record results of function evaluation for future use
merge -- merge two hash tables
merge(HashTable,HashTable,Function) -- merge two hash tables
mergePairs -- merge sorted lists of pairs
mergePairs(BasicList,BasicList,Function) -- merge sorted lists of pairs
method -- make a new method function
method(...,Binary=>...) -- make a new method function
method(...,Dispatch=>...) -- make a new method function
method(...,Options=>...) -- make a new method function
method(...,TypicalValue=>...) -- make a new method function
MethodFunction -- a type of method function
MethodFunctionBinary -- a type of method function
MethodFunctionSingle -- a type of method function
MethodFunctionWithOptions -- a type of method function
methodOptions -- recover the options used when a method function was created
methodOptions(Command) -- recover the options used when a method function was created
methodOptions(Function) -- recover the options used when a method function was created
methodOptions(ScriptedFunctor) -- recover the options used when a method function was created
methods -- list methods
methods for normal forms and remainder -- normal form of ring elements and matrices
methods(Command) -- list methods
methods(Manipulator) -- list methods
methods(Package) -- list methods
methods(ScriptedFunctor) -- list methods
methods(Sequence) -- list methods
methods(Symbol) -- list methods
methods(Thing) -- list methods
methods(Type) -- list methods
Michael E. Stillman
midpoint -- midpoint of an interval
min -- yields the minimum element in a list or sequence
min(GradedModule)
min(VisibleList) -- yields the minimum element in a list or sequence
minExponent
mingens -- minimal generator matrix
mingens(...,Strategy=>...) -- a Strategy option value
mingens(GroebnerBasis) -- (partially constructed) minimal generator matrix
mingens(Ideal) -- minimal generator matrix
mingens(Module) -- minimal generator matrix
mingle -- mingle elements of several lists
mingle(BasicList) -- mingle elements of several lists
minimal presentations and generators
minimalBetti -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
minimalBetti(...,DegreeLimit=>...) -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
minimalBetti(...,LengthLimit=>...) -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
minimalBetti(...,Weights=>...) -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
minimalBetti(Ideal) -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
minimalBetti(Module) -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
minimalBetti(MonomialIdeal) -- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
MinimalGenerators -- whether to compute minimal generators and return a trimmed set of generators
MinimalMatrix -- an optional argument
minimalPresentation -- compute a minimal presentation
minimalPresentation(...,Exclude=>...) -- compute a minimal presentation of a quotient ring
minimalPresentation(ChainComplex) -- minimal presentation of a module
minimalPresentation(ChainComplexMap) -- minimal presentation of a module
minimalPresentation(CoherentSheaf) -- minimal presentation of a module
minimalPresentation(GradedModule) -- minimal presentation of a module
minimalPresentation(GradedModuleMap) -- minimal presentation of a module
minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
minimalPresentation(Matrix) -- minimally present source and target of a matrix
minimalPresentation(Module) -- minimal presentation of a module
minimalPresentation(Ring) -- compute a minimal presentation of a quotient ring
minimalPresentationMap -- compute a minimal presentation of a quotient ring
minimalPresentationMapInv -- compute a minimal presentation of a quotient ring
Minimize -- an optional argument
minimizeFilename -- minimize a file name
MinimumVersion -- the minimum required version of a program
minors -- ideal generated by minors
minors(...,First=>...) -- ideal generated by minors
minors(...,Limit=>...) -- ideal generated by minors
minors(...,Strategy=>...) -- choose between Bareiss, Cofactor and Dynamic algorithms
minors(ZZ,Matrix) -- ideal generated by minors
minPosition -- position of the smallest element
minPosition(BasicList) -- position of the smallest element
Minus -- the class of all minus expressions
minus -- additive inverse
mkdir
mod -- reduce modulo an integer
Module -- the class of all modules
module -- make or get a module
Module ** ChainComplex -- tensor product
Module ** ChainComplexMap -- tensor product
Module ** GradedModule -- a binary operator, usually used for tensor product or Cartesian product
Module ** Matrix -- tensor product
Module ** Module -- tensor product
Module ** Ring -- tensor product
Module + Module -- sum of submodules
Module ++ GradedModule -- direct sum of modules
Module ++ Module -- direct sum of modules
Module / Ideal -- quotient module
Module / List -- quotient module
Module / Module -- quotient module
Module / RingElement -- quotient module
Module / Sequence -- quotient module
Module / Vector -- quotient module
Module == Ideal -- equality
Module == Module -- equality
Module == ZZ -- equality
Module ^ Array -- projection onto summand
Module ^ List -- projection onto summand
Module ^ ZZ -- direct sum
Module ^** ZZ -- tensor power
Module _ Array -- inclusion from summand
Module _ List -- map from free module to some generators
Module _ ZZ
Module ~ -- make a coherent sheaf
Module Array -- make a chain complex from a module
module homomorphisms
module(CoherentSheaf) -- get the module defining a coherent sheaf
module(Ring) -- make or get a module
module(SheafOfRings) -- make or get a module
module(Vector) -- the module of a vector
modules
modules in Macaulay2
modulo -- find the pre-image (pullback) of image of a map (low level version)
modulo(Matrix,Matrix) -- find the pre-image (pullback) of image of a map (low level version)
modulo(Matrix,Nothing) -- find the pre-image (pullback) of image of a map (low level version)
modulo(Nothing,Matrix) -- find the pre-image (pullback) of image of a map (low level version)
Monoid -- the class of all monoids
monoid -- make or retrieve a monoid
Monoid ** Monoid -- tensor product of monoids
monoid(...,DegreeLift=>...) -- specify maps between degree groups
monoid(...,DegreeMap=>...) -- specify maps between degree groups
monoid(...,DegreeRank=>...) -- specify the degrees of the variables
monoid(...,Degrees=>...) -- specify the degrees of the variables
monoid(...,Global=>...) -- specify local or global monomial order
monoid(...,Heft=>...) -- specify a heft vector
monoid(...,Inverses=>...) -- allow negative exponents in monomials
monoid(...,Join=>...) -- specify how to handle degrees in the coefficient ring
monoid(...,Local=>...) -- specify local or global monomial order
monoid(...,MonomialOrder=>...) -- specify the monomial ordering
monoid(...,MonomialSize=>...) -- specify the bit-length of monomial exponents in the ring
monoid(...,SkewCommutative=>...) -- specify Skew commuting variables in the ring
monoid(...,VariableBaseName=>...) -- specify the names of the indeterminates
monoid(...,Variables=>...) -- specify the names of the indeterminates
monoid(...,Weights=>...) -- specify weights of the variables
monoid(...,WeylAlgebra=>...) -- specify differential operators in the ring
monoid(Array) -- make or retrieve a monoid
monoid(List) -- make or retrieve a monoid
monoid(Ring) -- make or retrieve a monoid
MonoidElement -- the class of all monoid elements
MonoidElement == MonoidElement -- equality
Monomial -- value of an optional argument
monomial orderings
monomial orders for free modules
monomialCurveIdeal -- make the ideal of a monomial curve
MonomialIdeal -- the class of all monomial ideals handled by the engine
monomialIdeal -- make a monomial ideal
MonomialIdeal * Ideal -- product of ideals
MonomialIdeal * Module -- a binary operator, usually used for multiplication
MonomialIdeal * MonomialIdeal -- a binary operator, usually used for multiplication
MonomialIdeal * Ring -- a binary operator, usually used for multiplication
MonomialIdeal + Ideal -- sum of ideals
MonomialIdeal + MonomialIdeal -- a unary or binary operator, usually used for addition
MonomialIdeal - MonomialIdeal -- monomial ideal difference
MonomialIdeal == Ideal -- equality
MonomialIdeal == MonomialIdeal -- equality
MonomialIdeal == Ring -- equality
MonomialIdeal == ZZ -- equality
MonomialIdeal ^ Array -- bracket power of an ideal
MonomialIdeal ^ ZZ -- power of an ideal
MonomialIdeal _ ZZ
monomialIdeal(Ideal) -- monomial ideal of lead monomials of a Gröbner basis
monomialIdeal(List) -- monomial ideal of lead monomials
monomialIdeal(Matrix) -- monomial ideal of lead monomials
monomialIdeal(Module) -- monomial ideal of lead monomials of a Gröbner basis
monomialIdeal(MonomialIdeal) -- monomial ideal of lead monomials of a Gröbner basis
monomialIdeal(RingElement) -- monomial ideal of lead monomials
monomialIdeal(Sequence) -- monomial ideal of lead monomials
MonomialOrder -- monomial ordering
Monomials -- whether to display the number of monomial
monomials -- matrix of monomials in a ring element or matrix
monomials(...,Variables=>...) -- specify variables
monomials(Matrix) -- matrix of monomials in a ring element or matrix
monomials(RingElement) -- matrix of monomials in a ring element or matrix
MonomialSize -- an optional argument
monomialSubideal -- find the largest monomial ideal in an ideal
monomialSubideal(Ideal) -- find the largest monomial ideal in an ideal
moveFile
moveFile(...,Verbose=>...)
moveFile(String)
moveFile(String,String)
moving or copying the Macaulay2 files
MPFI
MPFR
MPIR
MPSolve -- a library for finding roots of univariate polynomials
multidegree -- multidegree
multidegree(Ideal) -- multidegree
multidegree(Module) -- multidegree
multidegree(Ring) -- multidegree
multigraded -- convert a Betti tally into a multigraded Betti tally
multigraded(BettiTally) -- convert a Betti tally into a multigraded Betti tally
MultigradedBettiTally -- the class of all multigraded Betti tallies
MultigradedBettiTally Sequence -- the class of all multigraded Betti tallies
mutable -- whether something may be modified
mutable matrices
mutable(Thing) -- whether something may be modified
MutableHashTable -- the class of all mutable hash tables
mutableIdentity -- make a mutable identity matrix
mutableIdentity(...,Dense=>...) -- make a mutable identity matrix
mutableIdentity(Ring,ZZ) -- make a mutable identity matrix
mutableIdentity(RingFamily,ZZ) -- make a mutable identity matrix
MutableList -- the class of all mutable lists
MutableMatrix -- the class of all mutable matrices
mutableMatrix -- make a mutable matrix
MutableMatrix * MutableMatrix -- a binary operator, usually used for multiplication
MutableMatrix + MutableMatrix -- a unary or binary operator, usually used for addition
MutableMatrix - MutableMatrix -- a unary or binary operator, usually used for negation or subtraction
MutableMatrix == MutableMatrix -- equality
MutableMatrix == ZZ -- equality
MutableMatrix _ Sequence -- get entry of matrix
MutableMatrix _ Sequence = Thing -- assignment to an element of a mutable matrix
mutableMatrix(...,Dense=>...) -- make a mutable matrix
mutableMatrix(List) -- make a mutable matrix
mutableMatrix(Matrix) -- make a mutable matrix
mutableMatrix(MutableMatrix) -- make a mutable matrix
mutableMatrix(Ring,ZZ,ZZ) -- make a mutable matrix filled with zeroes
mutableMatrix(RingFamily,ZZ,ZZ) -- make a mutable matrix filled with zeroes
Name -- the preamble of a package
nanosleep -- sleep for a given number of nanoseconds
nauty
NCLex -- Non-commutative lexicographical order.
needs -- read Macaulay2 commands if necessary
needsPackage -- load a package if not already loaded
needsPackage(...,Configuration=>...) -- load a package if not already loaded
needsPackage(...,DebuggingMode=>...) -- load a package if not already loaded
needsPackage(...,FileName=>...) -- load a package if not already loaded
needsPackage(...,LoadDocumentation=>...) -- load a package if not already loaded
needsPackage(...,Reload=>...) -- load a package if not already loaded
needsPackage(String) -- load a package if not already loaded
Net -- the class of all nets and strings
net -- format for printing, as a net
Net == Net -- equality
Net == String -- equality
Net ^ ZZ -- raise a net or string
Net | Net -- join strings or nets
Net || Net -- join nets or strings vertically
net(String) -- convert a string to a net
net(Thing) -- format for printing, as a net
NetFile -- the class of all net files
NetFile << Manipulator -- the class of all net files
NetFile << Net -- the class of all net files
NetFile << String -- the class of all net files
netList -- a table of boxes
netList(...,Alignment=>...) -- a table of boxes
netList(...,BaseRow=>...) -- a table of boxes
netList(...,Boxes=>...) -- a table of boxes
netList(...,HorizontalSpace=>...) -- a table of boxes
netList(...,VerticalSpace=>...) -- a table of boxes
netList(VisibleList) -- a table of boxes
new -- new objects and new types
new ChainComplex -- make a new chain complex from scratch
new HashTable from List -- make a hash table from a list
new Manipulator from Function -- the class of all file manipulators
new URL from String -- a type representing a URL
newClass -- set the class and parent of an object
newCoordinateSystem -- change variables
newCoordinateSystem(PolynomialRing,Matrix) -- change variables
NewFromMethod
newline -- the new line character sequence
NewMethod
newNetFile -- create a new net file
NewOfFromMethod
NewOfMethod
newPackage -- the preamble of a package
newPackage(...,Authors=>...) -- the preamble of a package
newPackage(...,AuxiliaryFiles=>...) -- the preamble of a package
newPackage(...,CacheExampleOutput=>...) -- the preamble of a package
newPackage(...,Certification=>...) -- the preamble of a package
newPackage(...,Configuration=>...) -- the preamble of a package
newPackage(...,Date=>...) -- the preamble of a package
newPackage(...,DebuggingMode=>...) -- the preamble of a package
newPackage(...,Headline=>...) -- the preamble of a package
newPackage(...,HomePage=>...) -- the preamble of a package
newPackage(...,InfoDirSection=>...) -- the preamble of a package
newPackage(...,Keywords=>...) -- the preamble of a package
newPackage(...,OptionalComponentsPresent=>...) -- the preamble of a package
newPackage(...,PackageExports=>...) -- the preamble of a package
newPackage(...,PackageImports=>...) -- the preamble of a package
newPackage(...,Reload=>...) -- the preamble of a package
newPackage(...,UseCachedExampleOutput=>...) -- the preamble of a package
newPackage(...,Version=>...) -- the preamble of a package
newPackage(String) -- the preamble of a package
newRing -- make a copy of a ring, with some features changed
newRing(...,DegreeLift=>...) -- make a copy of a ring, with some features changed
newRing(...,DegreeMap=>...) -- make a copy of a ring, with some features changed
newRing(...,DegreeRank=>...) -- make a copy of a ring, with some features changed
newRing(...,Degrees=>...) -- make a copy of a ring, with some features changed
newRing(...,Global=>...) -- make a copy of a ring, with some features changed
newRing(...,Heft=>...) -- make a copy of a ring, with some features changed
newRing(...,Inverses=>...) -- make a copy of a ring, with some features changed
newRing(...,Join=>...) -- make a copy of a ring, with some features changed
newRing(...,Local=>...) -- make a copy of a ring, with some features changed
newRing(...,MonomialOrder=>...) -- make a copy of a ring, with some features changed
newRing(...,MonomialSize=>...) -- make a copy of a ring, with some features changed
newRing(...,SkewCommutative=>...) -- make a copy of a ring, with some features changed
newRing(...,VariableBaseName=>...) -- make a copy of a ring, with some features changed
newRing(...,Variables=>...) -- make a copy of a ring, with some features changed
newRing(...,Weights=>...) -- make a copy of a ring, with some features changed
newRing(...,WeylAlgebra=>...) -- make a copy of a ring, with some features changed
newRing(PolynomialRing) -- make a copy of a ring, with some features changed
newRing(QuotientRing) -- make a copy of a ring, with some features changed
next -- get the next object from an iterator
next(Iterator) -- get the next object from an iterator
nextkey -- the next key in a database
nextPrime -- compute the smallest prime greater than or equal to a given number
nextPrime(Number) -- compute the smallest prime greater than or equal to a given number
NonAssociativeProduct -- the class of all nonassociative product expressions
NoPrint -- top level method for non-printing results
norm
norm(InexactField,MutableMatrix)
norm(InfiniteNumber,Matrix)
norm(InfiniteNumber,Number)
norm(InfiniteNumber,RingElement)
norm(List)
norm(Matrix)
norm(MutableMatrix)
norm(Number)
norm(RingElement)
norm(RR,Matrix)
norm(RR,MutableMatrix)
norm(RR,Number)
norm(RR,RingElement)
norm(Vector)
normal forms
normaliz
not -- negation
not documented yet
Nothing -- the empty class
Nothing << Manipulator -- print to a file
Nothing << Thing -- print to a file
Nothing == Nothing -- equality
notify -- whether to notify the user when a file is loaded
notImplemented -- print an 'not implemented' error message
NTL library
null -- the unique member of the empty class
nullaryMethods
nullhomotopy -- make a null homotopy
nullhomotopy(ChainComplexMap) -- make a null homotopy
nullSpace -- find the null space of a mutable matrix
nullSpace(MutableMatrix) -- find the null space of a mutable matrix
Number -- the class of all numbers
number -- count how many elements of a list satisfy a condition
Number % GroebnerBasis -- a binary operator, usually used for remainder and reduction
Number % Ideal -- a binary operator, usually used for remainder and reduction
Number % RingElement -- a binary operator, usually used for remainder and reduction
Number * Constant -- a binary operator, usually used for multiplication
Number * Matrix -- a binary operator, usually used for multiplication
Number + Constant -- a unary or binary operator, usually used for addition
Number + InfiniteNumber -- a unary or binary operator, usually used for addition
Number + Matrix -- a unary or binary operator, usually used for addition
Number - Constant -- a unary or binary operator, usually used for negation or subtraction
Number - InfiniteNumber -- a unary or binary operator, usually used for negation or subtraction
Number - Matrix -- a unary or binary operator, usually used for negation or subtraction
Number / Constant
Number // InfiniteNumber -- a binary operator, usually used for quotient
Number // Matrix -- a binary operator, usually used for quotient
Number // RingElement -- a binary operator, usually used for quotient
Number == InfiniteNumber -- equality
Number == Matrix -- equality
Number == RingElement -- equality
Number ^ Constant
Number ^ Ring -- lift to another ring
Number ^ RingFamily -- lift to another ring
Number _ Ring -- promote to another ring
numbered variables
NumberedVerticalList -- a type of visible self-initializing list that prints vertically
numColumns -- number of columns in a matrix or mutable matrix
numColumns(Matrix) -- number of columns in a matrix or mutable matrix
numColumns(MutableMatrix) -- number of columns in a matrix or mutable matrix
numerator -- numerator of a fraction
numerator(Divide) -- numerator of a fraction
numeric -- convert to floating point
numeric types
numeric(CC) -- convert to floating point
numeric(Constant) -- convert to floating point
numeric(InfiniteNumber) -- convert to floating point
numeric(Matrix) -- convert to floating point
numeric(Number) -- convert to floating point
numeric(RR) -- convert to floating point
numeric(RRi) -- convert to floating point
numeric(Vector) -- convert to floating point
numeric(VisibleList) -- convert to floating point
numeric(ZZ,CC) -- convert to floating point
numeric(ZZ,Constant) -- convert to floating point
numeric(ZZ,InfiniteNumber) -- convert to floating point
numeric(ZZ,Matrix) -- convert to floating point
numeric(ZZ,Number) -- convert to floating point
numeric(ZZ,RRi) -- convert to floating point
numeric(ZZ,Vector) -- convert to floating point
numeric(ZZ,VisibleList) -- convert to floating point
numericInterval -- converts to an interval
numericInterval(Constant) -- converts to an interval
numericInterval(ZZ,Constant) -- converts to an interval
numgens -- the number of generators
numgens(CoherentSheaf) -- the number of generators of the underlying module
numgens(EngineRing) -- number of generators of a polynomial ring
numgens(FractionField) -- number of generators of a polynomial ring
numgens(Ideal) -- number of generators of an ideal
numgens(Module) -- number of generators of a module
numgens(Monoid) -- number of generators of a monoid
numgens(MonomialIdeal) -- number of generators of a polynomial ring
numgens(PolynomialRing) -- number of generators of a polynomial ring
numgens(QuotientRing) -- number of generators of a polynomial ring
numgens(Ring) -- number of generators of a polynomial ring
numRows -- number of rows in a matrix or mutable matrix
numRows(Matrix) -- number of rows in a matrix or mutable matrix
numRows(MutableMatrix) -- number of rows in a matrix or mutable matrix
obtaining the monomial order of a ring
odd -- tell whether an integer is odd
of -- a keyword
ofClass -- English phrases for types
ofClass(ImmutableType) -- English phrases for types
ofClass(List) -- English phrases for types
ofClass(Type) -- English phrases for types
on -- trace a function each time it's run
on(...,CallLimit=>...) -- trace a function each time it's run
on(...,GenerateAssertions=>...) -- trace a function each time it's run
on(...,Name=>...) -- trace a function each time it's run
OneExpression -- the class all one expressions
OO -- the structure sheaf
oo -- the last output value
OO _ Variety -- the structure sheaf
ooo -- the next to the last output value
oooo -- the third to the last output value
openDatabase -- open a database file
openDatabase(String) -- open a database file
openDatabaseOut -- open a database file for writing
openDatabaseOut(String) -- open a database file for writing
openFiles -- list the open files
openIn -- open an input file
openIn(String) -- open an input file
openInOut -- open an input output file
openInOut(File) -- open an input output file
openInOut(String) -- open an input output file
openListener -- open a port for listening
openListener(String) -- open a port for listening
openOut -- open an output file
openOut(String) -- open an output file
openOutAppend -- open an output file for appending
openOutAppend(String) -- open an output file for appending
operatorAttributes -- a hashtable with information about Macaulay2 operators
operators
Option -- the class of all key-value pairs K => V
Option ++ Option -- direct sum of modules or maps
optional argument documentation template
optional argument name documentation template
OptionalComponentsPresent -- an optional argument
Options -- an optional argument
options -- get options
options(Command) -- get optional arguments and default values of functions and methods
options(Function) -- get optional arguments and default values of functions and methods
options(Monoid) -- get values used for optional arguments
options(Package)
options(PolynomialRing) -- get values used for optional arguments
options(QuotientRing) -- get values used for optional arguments
options(Ring) -- get values used for optional arguments
options(Sequence) -- get optional arguments and default values of functions and methods
options(ZZ) -- get optional arguments and default values of functions and methods
OptionTable -- the class of hash tables for optional arguments
OptionTable ++ List -- a binary operator, usually used for direct sum
OptionTable ++ OptionTable -- a binary operator, usually used for direct sum
OptionTable >> Function -- attaching options to a function
or -- disjunction
Order -- specify the order of a Hilbert series required
order -- a key used internally
OrderedMonoid -- the class of all ordered monoids
Other sources of information about Macaulay2
OutputDictionary -- the dictionary for output values
Outputs -- outputs for a function in a documentation node
override -- override default values for optional arguments
overview documentation template
pack -- pack elements of a list or string into several shorter lists or strings
pack(BasicList,ZZ) -- pack elements of a list or string into several shorter lists or strings
pack(String,ZZ) -- pack elements of a list or string into several shorter lists or strings
pack(ZZ,BasicList) -- pack elements of a list or string into several shorter lists or strings
pack(ZZ,String) -- pack elements of a list or string into several shorter lists or strings
Package -- the class of all packages
package -- get containing package
package documentation template
package(Array) -- get containing package
package(Dictionary) -- get containing package
package(DocumentTag) -- get containing package
package(Function) -- get containing package
package(HashTable) -- get containing package
package(Nothing) -- get containing package
package(Option) -- get containing package
package(Package) -- get containing package
package(Sequence) -- get containing package
package(String) -- get containing package
package(Symbol) -- get containing package
package(Thing) -- get containing package
PackageDictionary -- the dictionary for names of packages
PackageExports -- an optional argument
PackageImports -- an optional argument
packages -- Macaulay2 packages
packages provided with Macaulay2
packing monomials for efficiency
pad -- pad a string or net with spaces
pad(Net,ZZ) -- pad a string or net with spaces
pad(String,ZZ) -- pad a string or net with spaces
pad(ZZ,Net) -- pad a string or net with spaces
pad(ZZ,String) -- pad a string or net with spaces
pager -- display with paging
PairLimit -- an optional argument
pairs -- list the pairs in a hash table, dictionary, or basic list
pairs(BasicList) -- list the pairs in a hash table, dictionary, or basic list
pairs(Dictionary) -- list the pairs in a hash table, dictionary, or basic list
pairs(HashTable) -- list the pairs in a hash table, dictionary, or basic list
PairsRemaining -- whether to display the number of S-pairs remaining
parallel programming with threads and tasks
parent -- parent type of an object
Parenthesize
parsing precedence, in detail
part -- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,InfiniteNumber,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,InfiniteNumber,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,ZZ,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(InfiniteNumber,ZZ,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(List,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,Nothing,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,Nothing,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,ZZ,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(Nothing,ZZ,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,InfiniteNumber,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,InfiniteNumber,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,Nothing,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,Nothing,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,ZZ,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
part(ZZ,ZZ,VisibleList,RingElement) -- select terms of a polynomial by degree(s) or weight(s)
Partition -- a type of list representing a partition of a natural number
partition -- partition a set or list by values of a function
Partition _ ZZ -- get element
partition(Function,VirtualTally) -- partition a set or list by values of a function
partition(Function,VirtualTally,VisibleList) -- partition a set or list by values of a function
partition(Function,VisibleList) -- partition a set or list by values of a function
partition(Function,VisibleList,VisibleList) -- partition a set or list by values of a function
partitions -- list the partitions of an integer
partitions(ZZ) -- list the partitions of an integer
partitions(ZZ,ZZ) -- list the partitions of an integer
parts -- display terms of a polynomial degree by degree
parts(RingElement) -- display terms of a polynomial degree by degree
path -- list of directories to look in
pdim -- calculate the projective dimension
pdim(BettiTally) -- the class of all Betti tallies
pdim(CoherentSheaf) -- calculate the projective dimension
pdim(Module) -- calculate the projective dimension of a module
peek -- examine contents of an object
peek' -- examine contents of an object
peek'(ZZ,Thing) -- examine contents of an object
permanents -- ideal generated by square permanents of a matrix
permanents(ZZ,Matrix) -- ideal generated by square permanents of a matrix
permutations -- produce all permutations of a list
permutations(VisibleList) -- produce all permutations of a list
permutations(ZZ) -- produce all permutations of a list
Pfaffians
pfaffians -- ideal generated by Pfaffians
pfaffians(ZZ,Matrix) -- ideal generated by Pfaffians
pi
pivots -- list of pivot locations of a matrix
pivots(Matrix) -- list of pivot locations of a matrix
plus -- addition
poincare -- assemble degrees of a ring, module, or ideal into a polynomial
poincare(BettiTally) -- the class of all Betti tallies
poincare(ChainComplex) -- assemble degrees of a chain complex into a polynomial
poincare(Ideal) -- assemble degrees of a ring, module, or ideal into a polynomial
poincare(Module) -- assemble degrees of a ring, module, or ideal into a polynomial
poincare(MonomialIdeal) -- assemble degrees of a ring, module, or ideal into a polynomial
poincare(Ring) -- assemble degrees of a ring, module, or ideal into a polynomial
poincareN -- assemble degrees into polynomial
poincareN(ChainComplex) -- assemble degrees into polynomial
polarize -- given a monomial ideal, computes the squarefree monomial ideal obtained via polarization
polarize(...,VariableBaseName=>...) -- given a monomial ideal, computes the squarefree monomial ideal obtained via polarization
polarize(MonomialIdeal) -- given a monomial ideal, computes the squarefree monomial ideal obtained via polarization
polynomial rings
PolynomialRing -- the class of all ordered monoid rings
Position -- monomial ordering
position -- the first element of a list satisfying a condition
position(...,Reverse=>...) -- the first element of a list satisfying a condition
position(VisibleList,Function) -- the first element of a list satisfying a condition
position(VisibleList,VisibleList,Function) -- the first element of a list satisfying a condition
positions -- which elements of a list satisfy a condition
positions(MutableList,Function) -- which elements of a list satisfy a condition
positions(VisibleList,Function) -- which elements of a list satisfy a condition
POSIX -- evaluate a regular expression search
Postfix -- a hashtable with information about Macaulay2 operators
Power -- the class of all power expressions
power -- power
powermod -- powers of integers mod N
powermod(ZZ,ZZ,ZZ) -- powers of integers mod N
powers
precedence of operators
Precision -- name of an optional argument.
precision
precision(FractionField)
precision(GaloisField)
precision(InexactField)
precision(InexactNumber)
precision(Matrix)
precision(MutableMatrix)
precision(Number)
precision(PolynomialRing)
precision(QuotientRing)
precision(Ring)
precision(RingElement)
Prefix -- a hashtable with information about Macaulay2 operators
prefixDirectory -- the prefix directory
prefixPath -- absolute locations of Macaulay2 files
preimage -- preimage of an ideal under a ring map
preimage(RingMap,Ideal) -- preimage of an ideal under a ring map
prepend -- add an element to the beginning of a list
prepend(Thing,BasicList) -- add an element to the beginning of a list
presentation -- presentation of a module or ring
presentation(Module) -- presentation of a module
presentation(PolynomialRing,QuotientRing) -- presentation of a quotient ring
presentation(QuotientRing) -- presentation of a quotient ring
pretty -- a pretty printer
PrimitiveElement -- specify a primitive element
Print -- top level method for printing results
print -- print something
printerr -- print something to stderr
printing and formatting for new classes
printing to the screen
printingAccuracy -- current accuracy for printing numbers
printingLeadLimit -- maximum number of leading zeroes to use when printing real numbers
printingPrecision -- current precision for printing numbers
printingSeparator -- string used to separate mantissa from exponent when printing real numbers
printingTimeLimit
printingTrailLimit -- maximum number of additional trailing digits to use when printing real numbers
printString -- lowlevel function to print a string, net, or symbol
printWidth
processID -- the process identifier
Product -- the class of all product expressions
product
product(List) -- product of elements
product(Set) -- product of elements
product(VirtualTally) -- product of elements
product(VisibleList,Function) -- product of values of a function
product(VisibleList,VisibleList,Function) -- product of results of applying a function pairwise
product(ZZ,Function) -- product of consecutive values of a function
ProductOrder -- product ordering
profile -- profile a function
profile(Function) -- profile a function
profile(String,Function) -- profile a function
profileSummary -- display profiling data
Program -- external program object
programPaths -- user-defined external program paths
ProgramRun -- result of running an external program
Proj -- make a projective variety
Proj(Ring) -- make a projective variety
projections, inclusions, and permutations
Projective -- whether to produce a projective Hilbert polynomial
ProjectiveHilbertPolynomial -- the class of all Hilbert polynomials
projectiveHilbertPolynomial -- Hilbert polynomial of projective space
ProjectiveHilbertPolynomial * ZZ -- a binary operator, usually used for multiplication
ProjectiveHilbertPolynomial + ProjectiveHilbertPolynomial -- a unary or binary operator, usually used for addition
ProjectiveHilbertPolynomial + ZZ -- a unary or binary operator, usually used for addition
ProjectiveHilbertPolynomial - ProjectiveHilbertPolynomial -- a unary or binary operator, usually used for negation or subtraction
ProjectiveHilbertPolynomial - ZZ -- a unary or binary operator, usually used for negation or subtraction
ProjectiveHilbertPolynomial == ProjectiveHilbertPolynomial -- equality
ProjectiveHilbertPolynomial == ZZ -- equality
ProjectiveHilbertPolynomial ZZ -- value of polynomial
projectiveHilbertPolynomial(ZZ) -- Hilbert polynomial of projective space
projectiveHilbertPolynomial(ZZ,ZZ) -- Hilbert polynomial of projective space
ProjectiveVariety -- the class of all projective varieties
promote -- promote to another ring
promote(RR,type of QQ) -- promote to another ring
protect -- protect a symbol
Prune -- value of an optional argument
prune -- prune, e.g., compute a minimal presentation
prune(...,Exclude=>...) -- compute a minimal presentation of a quotient ring
prune(ChainComplex) -- minimal presentation of a module
prune(ChainComplexMap) -- minimal presentation of a module
prune(CoherentSheaf) -- minimal presentation of a module
prune(GradedModule) -- minimal presentation of a module
prune(GradedModuleMap) -- minimal presentation of a module
prune(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal
prune(Matrix) -- minimally present source and target of a matrix
prune(Module) -- minimal presentation of a module
prune(Ring) -- compute a minimal presentation of a quotient ring
pruningMap
Pseudocode -- the class of pseudocodes
pseudocode -- produce the pseudocode for a function
pseudoRemainder -- compute the pseudo-remainder
pseudoRemainder(RingElement,RingElement) -- compute the pseudo-remainder
pushForward
pushForward(...,DegreeLimit=>...)
pushForward(...,MonomialOrder=>...)
pushForward(...,PairLimit=>...)
pushForward(...,StopBeforeComputation=>...)
pushForward(...,UseHilbertFunction=>...)
pushForward(RingMap,Module)
QQ -- the class of all rational numbers
QQ ! -- factorial
QQ % QQ -- a binary operator, usually used for remainder and reduction
QQ % ZZ -- a binary operator, usually used for remainder and reduction
QQ * BettiTally -- the class of all Betti tallies
QQ * CC -- a binary operator, usually used for multiplication
QQ * QQ -- a binary operator, usually used for multiplication
QQ * RR -- a binary operator, usually used for multiplication
QQ * RRi -- a binary operator, usually used for multiplication
QQ * ZZ -- a binary operator, usually used for multiplication
QQ + CC -- a unary or binary operator, usually used for addition
QQ + QQ -- a unary or binary operator, usually used for addition
QQ + RR -- a unary or binary operator, usually used for addition
QQ + RRi -- a unary or binary operator, usually used for addition
QQ + ZZ -- a unary or binary operator, usually used for addition
QQ - CC -- a unary or binary operator, usually used for negation or subtraction
QQ - QQ -- a unary or binary operator, usually used for negation or subtraction
QQ - RR -- a unary or binary operator, usually used for negation or subtraction
QQ - RRi -- a unary or binary operator, usually used for negation or subtraction
QQ - ZZ -- a unary or binary operator, usually used for negation or subtraction
QQ / CC -- a binary operator, usually used for division
QQ / QQ -- a binary operator, usually used for division
QQ / RR -- a binary operator, usually used for division
QQ / RRi -- a binary operator, usually used for division
QQ / ZZ -- a binary operator, usually used for division
QQ // QQ -- a binary operator, usually used for quotient
QQ // ZZ -- a binary operator, usually used for quotient
QQ == CC -- equality
QQ == QQ -- equality
QQ == RR -- equality
QQ == RRi -- equality
QQ == ZZ -- equality
QRDecomposition -- compute a QR decomposition of a real matrix
QRDecomposition(Matrix) -- compute a QR decomposition of a real matrix
QRDecomposition(MutableMatrix) -- compute a QR decomposition of a real matrix
quit -- quit the program
Quotient -- value of an optional argument
quotient -- quotient or division
quotient rings
quotient' -- matrix quotient (opposite)
quotient'(Matrix,Matrix) -- matrix quotient (opposite)
quotient(Ideal) -- submodule to quotient module
quotient(Matrix,GroebnerBasis) -- matrix quotient
quotient(Matrix,Matrix) -- matrix quotient
quotient(Module) -- submodule to quotient module
quotientRemainder -- matrix quotient and remainder
quotientRemainder' -- matrix quotient and remainder (opposite)
quotientRemainder'(Matrix,Matrix) -- matrix quotient and remainder (opposite)
quotientRemainder(InexactNumber,RingElement) -- quotient and remainder
quotientRemainder(Matrix,GroebnerBasis) -- matrix quotient and remainder
quotientRemainder(Matrix,Matrix) -- matrix quotient and remainder
quotientRemainder(Number,RingElement) -- quotient and remainder
quotientRemainder(RingElement,InexactNumber) -- quotient and remainder
quotientRemainder(RingElement,Number) -- quotient and remainder
quotientRemainder(RingElement,RingElement) -- quotient and remainder
quotientRemainder(ZZ,ZZ) -- quotient and remainder
QuotientRing -- the class of all quotient rings
RaiseError -- whether to raise an error
random -- get a random object
random and generic matrices
random(...,Density=>...) -- make a random module map
random(...,Height=>...) -- get a random homogeneous element from a graded ring
random(...,MaximalRank=>...) -- make a random module map
random(...,UpperTriangular=>...) -- make a random module map
random(List) -- shuffle a list randomly
random(List,Ideal) -- get a random homogeneous element from a graded ideal
random(List,Ring) -- get a random homogeneous element from a graded ring
random(Module,Module) -- make a random module map
random(RR) -- get a random integer or real number
random(RR,RR) -- get a random integer or real number
random(Type) -- get a random object of a type
random(ZZ) -- get a random integer or real number
random(ZZ,Ideal) -- get a random homogeneous element from a graded ideal
random(ZZ,Ring) -- get a random homogeneous element from a graded ring
random(ZZ,ZZ) -- get a random integer or real number
randomKRationalPoint -- Pick a random K rational point on the scheme X defined by I
randomKRationalPoint(Ideal) -- Pick a random K rational point on the scheme X defined by I
randomMutableMatrix -- a random mutable matrix of integers
randomMutableMatrix(...,Dense=>...) -- a random mutable matrix of integers
randomMutableMatrix(ZZ,ZZ,RR,ZZ) -- a random mutable matrix of integers
Range -- can be assigned a integral interval
rank -- compute the rank
rank of a matrix
rank(CoherentSheaf) -- compute the rank
rank(GradedModule) -- compute the rank
rank(Matrix) -- compute the rank
rank(Module) -- compute the rank
rank(MutableMatrix) -- compute the rank
read -- read from a file
read(File) -- read from a file
read(File,ZZ) -- read from a file
read(Sequence) -- read from a file
read(String) -- read from a file
readDirectory -- read the contents of a directory
reading files
reading the documentation
readlink -- readlink fn
readPackage -- read the package preamble
readPackage(...,FileName=>...) -- read the package preamble
readPackage(Package) -- read the package preamble
readPackage(String) -- read the package preamble
RealField -- the class of all real fields
realPart -- real part
realPart(Number) -- real part
realpath -- convert a filename to one passing through no symbolic links
recursionDepth -- the current recursion depth
recursionLimit -- set the limit on recursion
Reduce -- an optional argument
reducedRowEchelonForm -- compute the reduced row echelon form of a matrix or mutable matrix over a field
reducedRowEchelonForm(Matrix) -- compute the reduced row echelon form of a matrix or mutable matrix over a field
reducedRowEchelonForm(MutableMatrix) -- compute the reduced row echelon form of a matrix or mutable matrix over a field
reduceHilbert -- reduce a Hilbert series expression
reduceHilbert(Divide) -- reduce a Hilbert series expression
References -- references listed in a documentation node
regex -- evaluate a regular expression search
regex(...,POSIX=>...) -- evaluate a regular expression search
regex(String,String) -- evaluate a regular expression search
regex(String,ZZ,String) -- evaluate a regular expression search
regex(String,ZZ,ZZ,String) -- evaluate a regular expression search
regexQuote -- escape special characters in regular expressions
regexQuote(String) -- escape special characters in regular expressions
registerFinalizer -- register a string that will be displayed when an object is garbage collected
regular expressions -- syntax for regular expressions
regularity -- compute the Castelnuovo-Mumford regularity
regularity(...,Weights=>...) -- compute the Castelnuovo-Mumford regularity
regularity(BettiTally) -- the class of all Betti tallies
regularity(ChainComplex) -- compute the Castelnuovo-Mumford regularity
regularity(Ideal) -- compute the Castelnuovo-Mumford regularity
regularity(Module) -- compute the Castelnuovo-Mumford regularity
regularizedBeta -- regularized beta function
regularizedGamma -- upper regularized gamma function
relations -- the defining relations
relations(Module) -- the defining relations
relativizeFilename -- relativize a file name
Reload -- an optional argument
remainder -- matrix remainder
remainder' -- matrix quotient and remainder (opposite)
remainder'(Matrix,Matrix) -- matrix quotient and remainder (opposite)
remainder(Matrix,GroebnerBasis) -- matrix remainder
remainder(Matrix,Matrix) -- matrix remainder
RemakeAllDocumentation -- an optional argument
remove -- remove an entry from a mutable hash table, list, or database
remove(Database,String) -- remove an entry from a mutable hash table, list, or database
remove(HashTable,Thing) -- remove an entry from a mutable hash table, list, or database
remove(MutableList,ZZ) -- remove an entry from a mutable hash table, list, or database
removeDirectory -- remove a directory
removeFile -- remove a file
reorganize -- reorganize a database file
replace -- replacement in strings and lists
replace(String,String,String) -- replace substrings matching a regular expression
replace(String,String,String,POSIX=>...) -- replace substrings matching a regular expression
replace(ZZ,Thing,VisibleList) -- copy a list, replacing an element
replacements for commands and scripts from Macaulay
RerunExamples -- an optional argument
reshape -- reshape a matrix
reshape(Module,Module,Matrix) -- reshape a matrix
Resolution -- the class of all resolution computations
resolution -- projective resolution
resolution(...,DegreeLimit=>...) -- compute only up to this degree
resolution(...,FastNonminimal=>...) -- compute a non-minimal graded free resolution
resolution(...,HardDegreeLimit=>...)
resolution(...,LengthLimit=>...) -- stop when the resolution reaches this length
resolution(...,PairLimit=>...) -- stop when this number of pairs has been handled
resolution(...,SortStrategy=>...)
resolution(...,StopBeforeComputation=>...) -- whether to stop the computation immediately
resolution(...,Strategy=>...)
resolution(...,SyzygyLimit=>...) -- stop when this number of syzygies is reached
resolution(Ideal) -- compute a projective resolution of (the quotient ring corresponding to) an ideal
resolution(Matrix) -- given a module map represented by a matrix, produce a comparison map between resolutions of its source and target
resolution(Module) -- compute a free resolution of a module
resolution(MonomialIdeal) -- compute a projective resolution of (the quotient ring corresponding to) an ideal
restart -- restart Macaulay2
Result -- an optional argument
return -- return from a function
returnCode
Reverse -- an optional argument
reverse -- reverse a list, sequence, or string
reverse(BasicList) -- reverse a list, sequence, or string
reverse(String) -- reverse a list, sequence, or string
RevLex -- reverse lexicographic ordering
Right -- value of an optional argument
right -- right endpoint of an interval
right modules or left modules?
right shift
Ring -- the class of all rings
ring -- get the associated ring of an object
Ring * Ideal -- a binary operator, usually used for multiplication
Ring * MonomialIdeal -- a binary operator, usually used for multiplication
Ring * RingElement -- a binary operator, usually used for multiplication
Ring * Vector -- a binary operator, usually used for multiplication
Ring ** Ideal -- tensor product
Ring ** Matrix -- tensor product
Ring ** Module -- tensor product
Ring ** Ring -- tensor product of monoids
Ring / Ideal -- make a quotient ring
Ring / List -- make a quotient ring
Ring / Module -- make a quotient ring
Ring / MonomialIdeal -- make a quotient ring
Ring / RingElement -- make a quotient ring
Ring / Sequence -- make a quotient ring
Ring / ZZ -- make a quotient ring
Ring == Ideal -- equality
Ring == MonomialIdeal -- equality
Ring == ZZ -- equality
Ring ^ BettiTally -- the class of all Betti tallies
Ring ^ List -- make a free module
Ring ^ ZZ -- make a free module
Ring _ List -- make a monomial from a list of exponents
Ring _ String -- get a ring variable by name
Ring _ ZZ -- get a ring variable by index
Ring ~ -- make a coherent sheaf of rings
Ring Array -- the standard way to make a polynomial ring
Ring List -- make a local polynomial ring
Ring Monoid -- make a polynomial ring
ring(CC) -- get the associated ring of an object
ring(ChainComplex) -- get the associated ring of an object
ring(ChainComplexMap) -- get the associated ring of an object
ring(CoherentSheaf) -- get the associated ring of an object
ring(GradedModule) -- get the associated ring of an object
ring(GradedModuleMap) -- get the associated ring of an object
ring(GroebnerBasis) -- get the associated ring of an object
ring(Ideal) -- get the associated ring of an object
ring(Matrix) -- get the associated ring of an object
ring(Module) -- get the associated ring of an object
ring(MonomialIdeal) -- get the associated ring of an object
ring(MutableMatrix) -- get the associated ring of an object
ring(Number) -- get the associated ring of an object
ring(Resolution) -- get the associated ring of an object
ring(RingElement) -- get the associated ring of an object
ring(RR) -- get the associated ring of an object
ring(RRi) -- get the associated ring of an object
ring(SheafOfRings) -- get the associated ring of an object
ring(Variety) -- get the associated ring of an object
ring(Vector) -- get the associated ring of an object
RingElement -- the class of all ring elements handled by the engine
RingElement % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
RingElement % Ideal -- normal form of ring elements and matrices
RingElement % Matrix -- normal form of ring elements and matrices
RingElement % MonomialIdeal -- normal form of ring elements and matrices
RingElement % Number -- a binary operator, usually used for remainder and reduction
RingElement % RingElement -- normal form of ring elements and matrices
RingElement * ChainComplexMap -- a binary operator, usually used for multiplication
RingElement * Constant
RingElement * GradedModuleMap -- a binary operator, usually used for multiplication
RingElement * Ideal -- a binary operator, usually used for multiplication
RingElement * Matrix -- a binary operator, usually used for multiplication
RingElement * Module -- a binary operator, usually used for multiplication
RingElement * MonomialIdeal -- a binary operator, usually used for multiplication
RingElement * MutableMatrix -- a binary operator, usually used for multiplication
RingElement * RingElement -- a binary operator, usually used for multiplication
RingElement * Vector -- a binary operator, usually used for multiplication
RingElement + ChainComplexMap -- a unary or binary operator, usually used for addition
RingElement + Constant
RingElement + GradedModuleMap -- a unary or binary operator, usually used for addition
RingElement + Matrix -- a unary or binary operator, usually used for addition
RingElement + RingElement -- a unary or binary operator, usually used for addition
RingElement ++ Matrix -- direct sum of maps
RingElement ++ RingElement -- direct sum of maps
RingElement - ChainComplexMap -- a unary or binary operator, usually used for negation or subtraction
RingElement - Constant
RingElement - GradedModuleMap -- a unary or binary operator, usually used for negation or subtraction
RingElement - Matrix -- a unary or binary operator, usually used for negation or subtraction
RingElement - RingElement -- a unary or binary operator, usually used for negation or subtraction
RingElement .. RingElement -- a sequence of consecutive generators of a polynomial ring
RingElement ..< RingElement -- a sequence of consecutive generators of a polynomial ring
RingElement / Constant
RingElement / RingElement -- fraction
RingElement // GroebnerBasis -- factor a map through another
RingElement // Matrix -- factor a map through another
RingElement // MonomialIdeal -- factor a map through another
RingElement // Number -- a binary operator, usually used for quotient
RingElement // RingElement -- factor a map through another
RingElement == ChainComplexMap -- equality
RingElement == GradedModuleMap -- equality
RingElement == Matrix -- equality
RingElement == Number -- equality
RingElement == RingElement -- equality
RingElement == ZZ -- equality
RingElement \\ Matrix -- factor a map through another
RingElement ^ Ring -- lift to another ring
RingElement ^ RingFamily -- lift to another ring
RingElement ^ ZZ -- power
RingElement _ Ring -- promote to another ring
RingElement | Matrix -- join matrices horizontally
RingElement | RingElement -- join matrices horizontally
RingElement || Matrix -- join matrices vertically
RingElement || RingElement -- join matrices vertically
RingElement Array -- substitution of variables
RingFamily
RingFamily _* -- a unary postfix operator, used for indicating pushforward maps
RingMap -- the class of all ring maps
RingMap * RingMap -- a binary operator, usually used for multiplication
RingMap ** Matrix -- tensor product via a ring map
RingMap ** Module -- tensor product via a ring map
RingMap == ZZ -- equality
RingMap \ List -- apply a function to elements of a list
RingMap ^ ZZ
RingMap ChainComplex -- apply a ring map
RingMap Ideal -- apply a ring map
RingMap Matrix -- apply a ring map
RingMap Module -- apply a ring map
RingMap RingElement -- apply a ring map
RingMap Vector -- apply a ring map
rings
rootPath
roots -- compute the roots of a polynomial
roots(RingElement) -- compute the roots of a polynomial
roots(RingElement,Precision=>...) -- compute the roots of a polynomial
roots(RingElement,Unique=>...) -- compute the roots of a polynomial
rootURI
rotate -- rotate a list
rotate(ZZ,VisibleList) -- rotate a list
round -- round a number
round(CC) -- round a number
round(QQ) -- round a number
round(RR) -- round a number
round(ZZ) -- round a number
round(ZZ,RR) -- round a number
row and column operations
rowAdd -- add a multiple of one row to another
rowAdd(MutableMatrix,ZZ,Number,ZZ) -- add a multiple of one row to another
rowAdd(MutableMatrix,ZZ,RingElement,ZZ) -- add a multiple of one row to another
RowExpression -- the class of all matrix expressions
rowMult -- multiply a row by a ring element
rowMult(MutableMatrix,ZZ,Number) -- multiply a row by a ring element
rowMult(MutableMatrix,ZZ,RingElement) -- multiply a row by a ring element
rowPermute -- permute some rows
rowPermute(MutableMatrix,ZZ,List) -- permute some rows
rowRankProfile -- find the row rank profile of a mutable matrix
rowRankProfile(MutableMatrix) -- find the row rank profile of a mutable matrix
rowSwap -- interchange rows
rowSwap(MutableMatrix,ZZ,ZZ) -- interchange rows
RR -- the class of all real numbers
RR ! -- factorial
RR % QQ -- a binary operator, usually used for remainder and reduction
RR % RR -- a binary operator, usually used for remainder and reduction
RR % ZZ -- a binary operator, usually used for remainder and reduction
RR * CC -- a binary operator, usually used for multiplication
RR * QQ -- a binary operator, usually used for multiplication
RR * RR -- a binary operator, usually used for multiplication
RR * RRi -- a binary operator, usually used for multiplication
RR * ZZ -- a binary operator, usually used for multiplication
RR + CC -- a unary or binary operator, usually used for addition
RR + QQ -- a unary or binary operator, usually used for addition
RR + RR -- a unary or binary operator, usually used for addition
RR + RRi -- a unary or binary operator, usually used for addition
RR + ZZ -- a unary or binary operator, usually used for addition
RR - CC -- a unary or binary operator, usually used for negation or subtraction
RR - QQ -- a unary or binary operator, usually used for negation or subtraction
RR - RR -- a unary or binary operator, usually used for negation or subtraction
RR - RRi -- a unary or binary operator, usually used for negation or subtraction
RR - ZZ -- a unary or binary operator, usually used for negation or subtraction
RR / CC -- a binary operator, usually used for division
RR / QQ -- a binary operator, usually used for division
RR / RR -- a binary operator, usually used for division
RR / RRi -- a binary operator, usually used for division
RR / ZZ -- a binary operator, usually used for division
RR // QQ -- a binary operator, usually used for quotient
RR // RR -- a binary operator, usually used for quotient
RR // ZZ -- a binary operator, usually used for quotient
RR << ZZ
RR == CC -- equality
RR == QQ -- equality
RR == RR -- equality
RR == RRi -- equality
RR == ZZ -- equality
RR >> ZZ
RR' -- the parent class of all rings of real numbers
RRi -- the class of all real intervals
RRi * QQ -- a binary operator, usually used for multiplication
RRi * RR -- a binary operator, usually used for multiplication
RRi * RRi -- a binary operator, usually used for multiplication
RRi * ZZ -- a binary operator, usually used for multiplication
RRi + QQ -- a unary or binary operator, usually used for addition
RRi + RR -- a unary or binary operator, usually used for addition
RRi + RRi -- a unary or binary operator, usually used for addition
RRi + ZZ -- a unary or binary operator, usually used for addition
RRi - QQ -- a unary or binary operator, usually used for negation or subtraction
RRi - RR -- a unary or binary operator, usually used for negation or subtraction
RRi - RRi -- a unary or binary operator, usually used for negation or subtraction
RRi - ZZ -- a unary or binary operator, usually used for negation or subtraction
RRi / QQ -- a binary operator, usually used for division
RRi / RR -- a binary operator, usually used for division
RRi / RRi -- a binary operator, usually used for division
RRi / ZZ -- a binary operator, usually used for division
RRi << ZZ
RRi == QQ -- equality
RRi == RR -- equality
RRi == RRi -- equality
RRi == ZZ -- equality
RRi >> ZZ
rsort -- sort a list or matrix in reverse order
rsort(...,DegreeOrder=>...) -- specify Ascending, Descending, or null
rsort(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
rsort(List) -- sort a list or matrix in reverse order
rsort(Matrix) -- sort a list or matrix in reverse order
run -- run an external command
RunDirectory -- the directory from which to run a program
RunExamples -- an optional argument
runHooks -- run the hook functions stored in an object
runHooks(MutableHashTable,Thing,Thing) -- run the hook functions stored in an object
runHooks(Sequence,Thing) -- run the hook functions stored in an object
runHooks(Symbol,Thing) -- run the hook functions stored in an object
runLengthEncode -- run length encoding
runLengthEncode(VisibleList) -- run length encoding
running Macaulay2 in a terminal window
running Macaulay2 in emacs
runProgram -- run an external program
runProgram(...,KeepFiles=>...) -- run an external program
runProgram(...,RaiseError=>...) -- run an external program
runProgram(...,RunDirectory=>...) -- run an external program
runProgram(...,Verbose=>...) -- run an external program
runProgram(Program,String) -- run an external program
runProgram(Program,String,String) -- run an external program
same -- whether everything in a list is the same
saving polynomials and matrices in files
scan -- apply a function to each element in a list or sequence
scan(BasicList,BasicList,Function) -- apply a function to each element in a list or sequence
scan(BasicList,Function) -- apply a function to each element in a list or sequence
scan(Thing,Function) -- apply a function to each element in a list or sequence
scan(ZZ,Function) -- apply a function to each element in a list or sequence
scanKeys -- apply a function to each key in a hash table or database
scanKeys(Database,Function) -- apply a function to each key in a hash table or database
scanKeys(HashTable,Function) -- apply a function to each key in a hash table or database
scanLines -- apply a function to each line of a file
scanLines(Function,List) -- apply a function to each line of a file
scanLines(Function,String) -- apply a function to each line of a file
scanPairs -- apply a function to the pairs in a hash table
scanPairs(HashTable,Function) -- apply a function to the pairs in a hash table
scanValues -- apply a function to each value in a hash table or database
scanValues(HashTable,Function) -- apply a function to each value in a hash table or database
schedule -- schedule a task for execution
schedule(Function) -- schedule a task for execution
schedule(Function,Thing) -- schedule a task for execution
schedule(Task) -- schedule a task for execution
Schreyer orders -- induced monomial order on a free module
schreyerOrder -- create or obtain free modules with Schreyer monomial orders
schreyerOrder(Matrix) -- create a matrix with the same entries whose source free module has a Schreyer monomial order
schreyerOrder(Module) -- obtain Schreyer order information
Schubert -- find the Plücker ideal of a Schubert variety
Schubert(...,CoefficientRing=>...) -- find the Plücker ideal of a Schubert variety
Schubert(...,Variable=>...) -- find the Plücker ideal of a Schubert variety
Schubert(ZZ,ZZ,VisibleList) -- find the Plücker ideal of a Schubert variety
scriptCommandLine -- the command line arguments to be used when running a script
ScriptedFunctor -- the class of all scripted functors
ScriptedFunctor ^ Thing -- the class of all scripted functors
ScriptedFunctor _ Thing -- the class of all scripted functors
ScriptedFunctor Thing -- the class of all scripted functors
searchPath -- search a path for a file
searchPath(List,String) -- search a path for a file
searchPath(String) -- search a path for a file
sec -- secant
sec(CC) -- secant
sec(RR) -- secant
sec(RRi) -- secant
sech -- hyperbolic secant
sech(CC) -- hyperbolic secant
sech(RR) -- hyperbolic secant
sech(RRi) -- hyperbolic secant
SeeAlso -- cross-references to other documentation nodes
seeParsing -- display parsing precedence table for Macaulay2 operators
select -- select from a list, hash table, or string
select(BasicList,Function) -- select elements from a list
select(BasicList,Type) -- select elements of a given type in a list
select(HashTable,Function) -- select part of a hash table
select(String,String) -- select and reformat substrings matching a regular expression
select(String,String,POSIX=>...) -- select and reformat substrings matching a regular expression
select(String,String,String) -- select and reformat substrings matching a regular expression
select(String,String,String,POSIX=>...) -- select and reformat substrings matching a regular expression
select(Thing,Function) -- select elements from an object with an iterator
select(ZZ,BasicList,Function) -- select a limited number of elements from a list
select(ZZ,Function) -- select integers
select(ZZ,HashTable,Function) -- select a limited number of pairs from a hash table
selectInSubring -- select columns in a subring
selectInSubring(ZZ,Matrix) -- select columns in a subring
selectVariables -- make a subring of a polynomial ring generated by selected variables
selectVariables(List,PolynomialRing) -- make a subring of a polynomial ring generated by selected variables
SelfInitializingType -- the class of all self initializing types
SelfInitializingType \ VisibleList -- apply a function to elements of a list
SelfInitializingType \\ Thing -- apply a function
SelfInitializingType Thing -- the class of all self initializing types
separate -- split a string into substrings using a regular expression
separate(...,POSIX=>...) -- split a string into substrings using a regular expression
separate(String) -- split a string into substrings using a regular expression
separate(String,String) -- split a string into substrings using a regular expression
separate(String,ZZ,String) -- split a string into substrings using a regular expression
SeparateExec -- an optional argument
separateRegexp -- split a string into substrings using a regular expression
Sequence -- the class of all sequences -- (...)
sequence -- make a sequence
Sequence .. Sequence -- rectangular sequences of consecutive lists and sequences
Sequence ..< Sequence -- rectangular sequences of consecutive lists and sequences
Sequence <- Thing -- assignment with left side evaluated
Sequence == Sequence -- equality
Sequence | Sequence -- join lists, sequences or arrays
serialNumber -- serial number of a dictionary, task, symbol, mutable hash table, or mutable list,
Set -- the class of all sets
set -- make a set
Set #? Thing -- test set membership
Set * Set -- intersection of sets
Set ** Set -- Cartesian product
Set + Set -- set union
Set - List -- set difference
Set - Set -- set difference
set(VisibleList) -- make a set
setEcho -- turn on echoing
setGroupID -- set the process group identifier
setIOExclusive -- exclusive I/O for the current thread
setIOSynchronized -- synchronized I/O for threads
setIOUnSynchronized -- unsynchronized I/O for threads
setRandomSeed -- set starting point for random number generator
setRandomSeed(Sequence) -- set starting point for random number generator
setRandomSeed(String) -- set starting point for random number generator
setRandomSeed(ZZ) -- set starting point for random number generator
setting up the Macaulay2 emacs interface
setup
setup()
setupEmacs
setupEmacs()
sheaf -- make a coherent sheaf
sheaf(Module) -- make a coherent sheaf
sheaf(Ring) -- make a coherent sheaf of rings
sheaf(Variety) -- make a coherent sheaf
sheaf(Variety,Module) -- make a coherent sheaf
sheaf(Variety,Ring) -- make a coherent sheaf of rings
SheafExpression -- the class of sheaf expressions
sheafExt -- sheaf Ext of coherent sheaves
sheafExt^ZZ(CoherentSheaf,CoherentSheaf) -- sheaf Ext of coherent sheaves
sheafExt^ZZ(CoherentSheaf,SheafOfRings) -- sheaf Ext of coherent sheaves
sheafExt^ZZ(SheafOfRings,CoherentSheaf) -- sheaf Ext of coherent sheaves
sheafExt^ZZ(SheafOfRings,SheafOfRings) -- sheaf Ext of coherent sheaves
sheafHom -- sheaf Hom
sheafHom(CoherentSheaf,CoherentSheaf) -- sheaf Hom
sheafHom(CoherentSheaf,SheafOfRings) -- sheaf Hom
sheafHom(SheafOfRings,CoherentSheaf) -- sheaf Hom
sheafHom(SheafOfRings,SheafOfRings) -- sheaf Hom
SheafOfRings -- the class of sheaves of rings
SheafOfRings (*) -- sum of twists
SheafOfRings ^ List -- make a graded free coherent sheaf
SheafOfRings ^ ZZ -- direct sum
SheafOfRings LowerBound -- the class of lower bound objects
SheafOfRings ZZ -- canonical twist of a coherent sheaf
shield -- shield evaluation from interrupts
show -- display various objects in an external viewer
show(URL) -- display various objects in an external viewer
showClassStructure -- display class structure
showHtml -- display various objects in an external viewer
showStructure -- display parent structure
showTex -- display various objects in an external viewer
showUserStructure -- show parent structure for those types defined by user
simple Groebner basis computations over various rings
sin -- compute the sine
sin(CC) -- compute the sine
sin(RR) -- compute the sine
sin(RRi) -- compute the sine
Singular Book 1.1.8 -- computation in fields
Singular Book 1.1.9 -- computation in polynomial rings
Singular Book 1.1.10 -- methods for creating ring maps
Singular Book 1.2.3 -- leading data
Singular Book 1.2.13 -- monomial orderings
Singular Book 1.3.3 -- properties of ring maps
Singular Book 1.3.13 -- computation in quotient rings
Singular Book 1.3.15 -- computing with radicals
Singular Book 1.4.9 -- global versus local rings
Singular Book 1.5.10 -- realization of rings
Singular Book 1.6.13 -- normal form
Singular Book 1.7.10 -- standard bases
Singular Book 1.7.12 -- highest corner
Singular Book 1.8.1 -- ideal membership
Singular Book 1.8.2 -- linear combination of ideal members
Singular Book 1.8.4 -- elimination of variables
Singular Book 1.8.6 -- Zariski closure of the image
Singular Book 1.8.7 -- solving equations
Singular Book 1.8.9 -- radical membership
Singular Book 1.8.11 -- intersection of ideals
Singular Book 1.8.13 -- quotient of ideals
Singular Book 1.8.15 -- saturation
Singular Book 1.8.18 -- kernel of a ring map
Singular Book 1.8.19 -- algebraic dependence
Singular Book 1.8.20 -- subalgebra membership
Singular Book 2.1.6 -- matrix operations
Singular Book 2.1.7 -- maps induced by Hom
Singular Book 2.1.10 -- Submodules of A^n
Singular Book 2.1.13 -- kernel, image and cokernel of a module homomorphism
Singular Book 2.1.20 -- sum, intersection, module quotient
Singular Book 2.1.24 -- submodules, presentation of a module
Singular Book 2.1.26 -- computation of Hom
Singular Book 2.1.34 -- minimal presentations, prune
Singular Book 2.2.15 -- graded rings and modules
Singular Book 2.3.10 -- normal form
Singular Book 2.3.12 -- standard bases
Singular Book 2.4.12 -- resolution and Betti numbers
Singular Book 2.4.15 -- homogeneous resolution and graded Betti numbers
Singular Book 2.5.5 -- syzygies
Singular Book 2.5.18 -- Schreyer resolution
Singular Book 2.6.3 -- diagonal form
Singular Book 2.6.11 -- cyclic decomposition
Singular Book 2.6.15 -- Jordan normal form
Singular Book 2.7.5 -- tensor product of maps
Singular Book 2.7.9 -- tensor product of modules
Singular Book 2.7.14 -- tensor product of rings
Singular Book 2.8.1 -- module membership
Singular Book 2.8.3 -- elimination of module components
Singular Book 2.8.5 -- intersection of modules
Singular Book 2.8.6 -- quotient of modules
Singular Book 2.8.7 -- radical, zerodivisors of modules
Singular Book 2.8.8 -- annihilator and Fitting ideal
Singular Book 2.8.9 -- kernel of a module homomorphism
Singular Book 2.8.10 -- solving linear equations
Singular-Factory
singularLocus -- singular locus
singularLocus(AffineVariety) -- singular locus
singularLocus(Ideal) -- singular locus
singularLocus(ProjectiveVariety) -- singular locus
singularLocus(Ring) -- singular locus
sinh -- compute the hyperbolic sine
sinh(CC) -- compute the hyperbolic sine
sinh(RR) -- compute the hyperbolic sine
sinh(RRi) -- compute the hyperbolic sine
size -- the size of an object
size(RingElement) -- the size of an object
size2 -- number of binary digits to the left of the point
size2(CC) -- number of binary digits to the left of the point
size2(RR) -- number of binary digits to the left of the point
size2(RRi) -- number of binary digits to the left of the point
size2(ZZ) -- number of binary digits to the left of the point
SizeLimit -- an optional argument
SkewCommutative -- an optional argument
sleep -- sleep for a while
smithNormalForm -- smith normal form for a matrix over ZZ or a PID
smithNormalForm(...,ChangeMatrix=>...) -- smith normal form for a matrix over ZZ or a PID
smithNormalForm(...,KeepZeroes=>...) -- smith normal form for a matrix over ZZ or a PID
smithNormalForm(Matrix) -- smith normal form for a matrix over ZZ or a PID
solve -- solve linear equation(s)
solve(...,ClosestFit=>...) -- solve linear equation(s)
solve(...,MaximalRank=>...) -- solve linear equation(s)
solve(...,Precision=>...) -- solve linear equation(s)
solve(Matrix,Matrix) -- solve linear equation(s)
solve(MutableMatrix,MutableMatrix) -- solve linear equation(s)
someTerms -- select some terms of a polynomial
someTerms(RingElement,ZZ,ZZ) -- select some terms of a polynomial
Sort -- a Strategy option value
sort -- sort a list or columns of a matrix
sort(...,DegreeOrder=>...) -- specify Ascending, Descending, or null
sort(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
sort(List) -- sort a list
sort(Matrix) -- sort the columns of a matrix
sortColumns -- permutation giving sort order
sortColumns(...,DegreeOrder=>...) -- specify Ascending, Descending, or null
sortColumns(...,MonomialOrder=>...) -- specify Ascending or Descending monomial order
sortColumns(Matrix) -- permutation giving sort order
SortStrategy -- specify a strategy for sorting S-pairs
source -- source of a map
source(ChainComplexMap) -- find the source of a map of chain complexes
source(GradedModuleMap) -- find the source of a map of graded modules
source(Matrix) -- find the source module of matrix
source(RingMap) -- find the source ring for a map of rings
SourceCode -- sources displayed in a documentation node
SourceRing -- an optional argument
SPACE -- blank operator; often used for function application, making polynomial rings
span -- construct smallest interval
span(List) -- construct smallest interval
span(List,Precision=>...) -- construct smallest interval
span(Sequence) -- construct smallest interval
span(Sequence,Precision=>...) -- construct smallest interval
SparseMonomialVectorExpression -- the class of all sparse monomial vector expressions
SparseVectorExpression -- the class of all sparse vector expressions
Spec -- make an affine variety
Spec(Ring) -- make an affine variety
specifying typical values
splice -- remove subsequences from a sequence or list
splice(BasicList) -- remove subsequences from a sequence or list
splitWWW -- separate an http response into header and body
splitWWW(String) -- separate an http response into header and body
sqrt -- square root function
sqrt(CC) -- square root function
sqrt(RR) -- square root function
sqrt(RRi) -- square root function
stack -- join nets or string vertically
stack(BasicList) -- join nets or string vertically
Standard -- the standard top level printing method
standardForm -- convert to standard form
standardForm(MonoidElement) -- convert to standard form
standardForm(RingElement) -- convert to standard form
standardPairs -- find the standard pairs of a monomial ideal
standardPairs(MonomialIdeal) -- find the standard pairs of a monomial ideal
standardPairs(MonomialIdeal,List) -- find the standard pairs of a monomial ideal
stashValue -- stash values of functions in their arguments
status -- status of a resolution computation
status(...,Monomials=>...) -- status of a resolution computation
status(...,PairsRemaining=>...) -- status of a resolution computation
status(...,TotalPairs=>...) -- status of a resolution computation
status(ChainComplex) -- status of a resolution computation
status(GroebnerBasis) -- status of a resolution computation
status(Resolution) -- status of a resolution computation
stderr -- the standard error output file
stdio -- the standard input output file
step -- step by single lines in the debugger
StopBeforeComputation -- an optional argument
stopIfError -- whether to stop the program when an error occurs
StopIteration -- stop iteration
StopWithMinimalGenerators -- an optional argument
Strategy -- an optional argument
String -- the class of all strings
String # ZZ -- get value from list, hash table, database, or string
String #? ZZ -- check existence of value in a list, hash table, database, or string
String .. String -- a sequence of consecutive strings
String ..< String -- a sequence of consecutive strings, not including the endpoint
String / Command -- apply a function to elements of a list
String / Function -- apply a function to elements of a list
String << Thing -- print to a file
String == Net -- equality
String == String -- equality
String ^ Sequence -- vertically stacked copies of a string
String ^ ZZ -- raise a net or string
String _ Sequence -- substring
String _ ZZ -- get element from string
String | String -- join strings or nets
String | ZZ -- join strings or nets
strings and nets -- an overview of strings and nets in Macaulay2
sublists -- process interspersed subsequences of a visible list
sublists(VisibleList,Function) -- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Function) -- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Function,Function) -- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Function,Nothing) -- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Nothing) -- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Nothing,Function) -- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Nothing,Nothing) -- process interspersed subsequences of a visible list
submatrix
submatrix' -- exclude rows and/or columns of a matrix
submatrix'(Matrix,Nothing,VisibleList) -- exclude rows and/or columns of a matrix
submatrix'(Matrix,VisibleList) -- exclude rows and/or columns of a matrix
submatrix'(Matrix,VisibleList,Nothing) -- exclude rows and/or columns of a matrix
submatrix'(Matrix,VisibleList,VisibleList) -- exclude rows and/or columns of a matrix
submatrix(Matrix,VisibleList) -- select columns
submatrix(Matrix,VisibleList,VisibleList) -- select part of a matrix
submatrix(MutableMatrix,VisibleList) -- select columns
submatrix(MutableMatrix,VisibleList,VisibleList) -- select part of a matrix
submatrixByDegrees -- submatrix consisting of rows and columns in an interval or box of degrees
submatrixByDegrees(Matrix,List,List) -- submatrix consisting of rows and columns in an interval or box of degrees
submatrixByDegrees(Matrix,Sequence,Sequence) -- submatrix consisting of rows and columns in an interval or box of degrees
submatrixByDegrees(Matrix,ZZ,ZZ) -- submatrix consisting of rows and columns in an interval or box of degrees
submodules and quotients
Subnodes -- a menu of documentation nodes
subquotient -- make a subquotient module
subquotient modules -- the way Macaulay2 represents modules
subquotient(Matrix,Matrix) -- make a subquotient module
subquotient(Matrix,Nothing) -- make a subquotient module
subquotient(Module,Matrix,Matrix) -- make a subquotient module
subquotient(Module,Matrix,Nothing) -- make a subquotient module
subquotient(Module,Nothing,Matrix) -- make a subquotient module
subquotient(Module,Nothing,Nothing) -- make a subquotient module
subquotient(Nothing,Matrix) -- make a subquotient module
SubringLimit -- stop after finding enough elements of a subring
Subscript -- the class of all subscript expressions
subscript -- specify the function in a scripted functor for a subscript
subscripted variables
subsets -- produce the subsets of a set or list
subsets(List) -- produce the subsets of a set or list
subsets(List,ZZ) -- produce the subsets of a set or list
subsets(Sequence,ZZ) -- produce the subsets of a set or list
subsets(Set) -- produce the subsets of a set or list
subsets(Set,ZZ) -- produce the subsets of a set or list
subsets(ZZ) -- produce the subsets of a set or list
subsets(ZZ,ZZ) -- produce the subsets of a set or list
substitute -- substituting values for variables
substitute(Divide,Thing) -- substituting values for variables
substitute(Ideal,List) -- substituting values for variables
substitute(Ideal,Matrix) -- substituting values for variables
substitute(Ideal,Option) -- substituting values for variables
substitute(Ideal,Ring) -- substituting values for variables
substitute(Ideal,RingFamily) -- substituting values for variables
substitute(Matrix,List) -- substituting values for variables
substitute(Matrix,Matrix) -- substituting values for variables
substitute(Matrix,Option) -- substituting values for variables
substitute(Matrix,Ring) -- substituting values for variables
substitute(Matrix,RingFamily) -- substituting values for variables
substitute(Matrix,ZZ) -- substituting values for variables
substitute(Module,List) -- substituting values for variables
substitute(Module,Matrix) -- substituting values for variables
substitute(Module,Option) -- substituting values for variables
substitute(Module,Ring) -- substituting values for variables
substitute(Module,RingFamily) -- substituting values for variables
substitute(Number,Ring) -- substituting values for variables
substitute(Number,RingFamily) -- substituting values for variables
substitute(Power,Thing) -- substituting values for variables
substitute(Product,Thing) -- substituting values for variables
substitute(RingElement,List) -- substituting values for variables
substitute(RingElement,Matrix) -- substituting values for variables
substitute(RingElement,Option) -- substituting values for variables
substitute(RingElement,Ring) -- substituting values for variables
substitute(RingElement,RingFamily) -- substituting values for variables
substitute(Sum,Thing) -- substituting values for variables
substitute(Vector,List) -- substituting values for variables
substitute(Vector,Matrix) -- substituting values for variables
substitute(Vector,Option) -- substituting values for variables
substitute(Vector,Ring) -- substituting values for variables
substitute(Vector,RingFamily) -- substituting values for variables
substituting values for variables
substitution and maps between rings
substring -- extract part of a string
substring(Sequence,String) -- extract part of a string
substring(String,ZZ) -- extract part of a string
substring(String,ZZ,ZZ) -- extract part of a string
substring(ZZ,String) -- extract part of a string
substring(ZZ,ZZ,String) -- extract part of a string
subtable -- extract a subtable from a table
Sugarless -- compute a Gröbner basis
Sum -- the class of all sum expressions
sum -- compute the sum
sum(ChainComplex) -- direct sum of the components of a chain complex
sum(ChainComplexMap) -- direct sum of the components of a chain map
sum(List) -- sum the elements of a list
sum(Set) -- sum of elements
sum(VirtualTally) -- sum of elements
sum(VisibleList,Function) -- sum results of applying a function
sum(VisibleList,VisibleList,Function) -- sum results of applying a function pairwise
sum(ZZ,Function) -- sum consecutive values of a function
SumOfTwists -- the class of all sums of twists
sums, products, and powers of ideals
super -- get the ambient module
super(CoherentSheaf) -- get the ambient module
super(GradedModule) -- get the ambient module
super(Matrix) -- get the ambient module
super(Module) -- get the ambient module
super(Vector) -- get the ambient module
Superscript -- the class of all superscript expressions
superscript -- specify the function in a scripted functor for a superscript
support -- list of variables occurring in a polynomial or matrix
support(Ideal) -- list of variables occurring in the generators of an ideal
support(Matrix) -- list of variables occurring in a polynomial or matrix
support(RingElement) -- list of variables occurring in a polynomial or matrix
SVD -- singular value decomposition of a matrix
SVD(...,DivideConquer=>...) -- Use the lapack divide and conquer SVD algorithm
SVD(Matrix) -- singular value decomposition of a matrix
SVD(MutableMatrix) -- singular value decomposition of a matrix
switch -- copy a list, switching two elements
switch(ZZ,ZZ,VisibleList) -- copy a list, switching two elements
Symbol -- the class of all symbols
symbol -- get a symbol
Symbol .. Symbol -- sequences of consecutive symbols
Symbol ..< Symbol -- sequences of consecutive symbols
Symbol == Symbol -- equality
Symbol ? IndexedVariable -- comparison operator
Symbol _ Ring -- get a ring variable by name
Symbol _ Thing -- index variable
Symbol _ Thing = Thing -- assignment to an indexed variable
SymbolBody -- symbol bodies
symbolBody -- symbol bodies
symbols used as the name or value of an optional argument
symlinkDirectory -- make symbolic links for all files in a directory tree
symlinkDirectory(...,Exclude=>...) -- make symbolic links for all files in a directory tree
symlinkDirectory(...,FollowLinks=>...) -- make symbolic links for all files in a directory tree
symlinkDirectory(...,Undo=>...) -- make symbolic links for all files in a directory tree
symlinkDirectory(...,Verbose=>...) -- make symbolic links for all files in a directory tree
symlinkDirectory(String,String) -- make symbolic links for all files in a directory tree
symlinkFile -- make a symbolic link to a file
symmetric algebras
symmetricAlgebra -- the symmetric algebra of a module
symmetricAlgebra(...,DegreeLift=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,DegreeMap=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,DegreeRank=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Degrees=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Global=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Heft=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Inverses=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Join=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Local=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,MonomialOrder=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,MonomialSize=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,SkewCommutative=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,VariableBaseName=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Variables=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,Weights=>...) -- the symmetric algebra of a module
symmetricAlgebra(...,WeylAlgebra=>...) -- the symmetric algebra of a module
symmetricAlgebra(Matrix) -- the symmetric algebra of a module
symmetricAlgebra(Module) -- the symmetric algebra of a module
symmetricAlgebra(Nothing,Nothing,Matrix) -- the symmetric algebra of a module
symmetricAlgebra(Nothing,Ring,Matrix) -- the symmetric algebra of a module
symmetricAlgebra(Ring,Nothing,Matrix) -- the symmetric algebra of a module
symmetricAlgebra(Ring,Ring,Matrix) -- the symmetric algebra of a module
symmetricPower -- symmetric power
symmetricPower(ZZ,Matrix) -- symmetric power
symmetricPower(ZZ,Module) -- symmetric power
synonym -- synonym for members of a class
SYNOPSIS -- a standardized synopsis for use in documentation nodes
SYNOPSIS(...,BaseFunction=>...) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(...,Consequences=>...) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(...,Heading=>...) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(...,Inputs=>...) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(...,Outputs=>...) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(...,Usage=>...) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(List) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(Sequence) -- a standardized synopsis for use in documentation nodes
SYNOPSIS(Thing) -- a standardized synopsis for use in documentation nodes
system facilities
syz -- the syzygy matrix
syz(...,Algorithm=>...) -- compute the syzygy matrix
syz(...,BasisElementLimit=>...) -- compute the syzygy matrix
syz(...,DegreeLimit=>...) -- compute the syzygy matrix
syz(...,GBDegrees=>...) -- compute the syzygy matrix
syz(...,HardDegreeLimit=>...) -- compute the syzygy matrix
syz(...,MaxReductionCount=>...) -- compute the syzygy matrix
syz(...,PairLimit=>...) -- compute the syzygy matrix
syz(...,StopBeforeComputation=>...) -- compute the syzygy matrix
syz(...,Strategy=>...) -- compute the syzygy matrix
syz(...,SyzygyLimit=>...) -- compute the syzygy matrix
syz(...,SyzygyRows=>...) -- compute the syzygy matrix
syz(GroebnerBasis) -- retrieve the syzygy matrix
syz(Matrix) -- compute the syzygy matrix
Syzygies -- an optional argument
SyzygyLimit -- an optional argument
SyzygyMatrix -- an optional argument
SyzygyRows -- an optional argument
syzygyScheme -- construct a syzygy scheme
Table -- the class of all table expressions
table -- make a table from a binary function
take -- Take some elements from a list or sequence.
take(BasicList,List) -- Take some elements from a list or sequence.
take(BasicList,ZZ) -- Take some elements from a list or sequence.
take(Thing,ZZ) -- Take some elements from a list or sequence.
Tally -- the class of all tally results
tally -- tally the elements of a list, sequence, array, or string
Tally + Tally -- union of tallies
Tally - Tally -- difference of tallies
tally(String) -- tally the elements of a list, sequence, array, or string
tally(VisibleList) -- tally the elements of a list, sequence, array, or string
tan -- compute the tangent
tan(CC) -- compute the tangent
tan(RR) -- compute the tangent
tan(RRi) -- compute the tangent
tangentSheaf -- tangent sheaf of a projective variety
tangentSheaf(...,Minimize=>...) -- tangent sheaf of a projective variety
tangentSheaf(ProjectiveVariety) -- tangent sheaf of a projective variety
tanh -- compute the hyperbolic tangent
tanh(CC) -- compute the hyperbolic tangent
tanh(RR) -- compute the hyperbolic tangent
tanh(RRi) -- compute the hyperbolic tangent
target -- target of a map
target(ChainComplexMap) -- find the target of a map of chain complexes
target(GradedModuleMap) -- find the target of a map of graded modules
target(GroebnerBasis) -- find target of a Gröbner basis
target(Matrix) -- find the target module of matrix
target(RingMap) -- find the target ring for a map of rings
Task -- the class of all tasks
taskResult -- retrieve the value returned by a task
taskResult(Task) -- retrieve the value returned by a task
teaching emacs how to find M2
teaching emacs how to find M2-init.el
teaching M2 how to find its shared libraries
teaching your shell how to find M2
temporaryFileName -- make a temporary file name
tensor -- tensor product
tensor products of rings
tensor(List) -- tensor product
tensor(Monoid,Monoid) -- tensor product of monoids
tensor(Monoid,Monoid,DegreeLift=>...) -- tensor product of monoids
tensor(Monoid,Monoid,DegreeMap=>...) -- tensor product of monoids
tensor(Monoid,Monoid,DegreeRank=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Degrees=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Global=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Heft=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Inverses=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Join=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Local=>...) -- tensor product of monoids
tensor(Monoid,Monoid,MonomialOrder=>...) -- tensor product of monoids
tensor(Monoid,Monoid,MonomialSize=>...) -- tensor product of monoids
tensor(Monoid,Monoid,SkewCommutative=>...) -- tensor product of monoids
tensor(Monoid,Monoid,VariableBaseName=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Variables=>...) -- tensor product of monoids
tensor(Monoid,Monoid,Weights=>...) -- tensor product of monoids
tensor(Monoid,Monoid,WeylAlgebra=>...) -- tensor product of monoids
tensor(Ring,Ring) -- tensor product of monoids
tensor(Ring,Ring,DegreeLift=>...) -- tensor product of monoids
tensor(Ring,Ring,DegreeMap=>...) -- tensor product of monoids
tensor(Ring,Ring,DegreeRank=>...) -- tensor product of monoids
tensor(Ring,Ring,Degrees=>...) -- tensor product of monoids
tensor(Ring,Ring,Global=>...) -- tensor product of monoids
tensor(Ring,Ring,Heft=>...) -- tensor product of monoids
tensor(Ring,Ring,Inverses=>...) -- tensor product of monoids
tensor(Ring,Ring,Join=>...) -- tensor product of monoids
tensor(Ring,Ring,Local=>...) -- tensor product of monoids
tensor(Ring,Ring,MonomialOrder=>...) -- tensor product of monoids
tensor(Ring,Ring,MonomialSize=>...) -- tensor product of monoids
tensor(Ring,Ring,SkewCommutative=>...) -- tensor product of monoids
tensor(Ring,Ring,VariableBaseName=>...) -- tensor product of monoids
tensor(Ring,Ring,Variables=>...) -- tensor product of monoids
tensor(Ring,Ring,Weights=>...) -- tensor product of monoids
tensor(Ring,Ring,WeylAlgebra=>...) -- tensor product of monoids
tensor(RingMap,Matrix) -- tensor product via a ring map
tensor(RingMap,Module) -- tensor product via a ring map
tensor(Sequence) -- tensor product
tensorAssociativity -- associativity isomorphisms for tensor products
tensorAssociativity(ChainComplex,ChainComplex,ChainComplex) -- associativity isomorphisms for tensor products
tensorAssociativity(GradedModule,GradedModule,GradedModule) -- associativity isomorphisms for tensor products
tensorAssociativity(Module,Module,Module) -- associativity isomorphisms for tensor products
terms -- provide a list of terms of a polynomial
terms(Ring,RingElement) -- provide a list of terms of a polynomial
terms(RingElement) -- provide a list of terms of a polynomial
TEST -- add a test for a package
Test -- value of an optional argument
TEST(...,FileName=>...) -- add a test for a package
TEST(List) -- add a test for a package
TEST(String) -- add a test for a package
TestInput -- locate a package's tests
tests -- locate a package's tests
tests(Package) -- locate a package's tests
tests(String) -- locate a package's tests
tex -- convert to $\TeX$ format
TeXmacs -- the TeXmacs top level printing method
texMath -- convert to TeX math format
The authors
the debugger
The Macaulay2 language
then -- condition testing
Thing -- the class of all things
Thing != Expression -- the class of all expressions
Thing % Expression -- the class of all expressions
Thing & Expression -- the class of all expressions
Thing * Expression -- the class of all expressions
Thing * List -- a binary operator, usually used for multiplication
Thing ** Expression -- the class of all expressions
Thing + Expression -- the class of all expressions
Thing ++ Expression -- the class of all expressions
Thing - Expression -- the class of all expressions
Thing .. Expression -- the class of all expressions
Thing ..< Expression -- the class of all expressions
Thing / Expression -- the class of all expressions
Thing // Command -- apply a function
Thing // Function -- apply a function
Thing // SelfInitializingType -- apply a function
Thing : Expression -- the class of all expressions
Thing <== Expression -- the class of all expressions
Thing <=== Expression -- the class of all expressions
Thing <==> Expression -- the class of all expressions
Thing =!= Thing -- strict inequality
Thing == Expression -- the class of all expressions
Thing === Thing -- strict equality
Thing ===> Expression -- the class of all expressions
Thing ==> Expression -- the class of all expressions
Thing => Thing -- produce an Option
Thing @ Expression -- the class of all expressions
Thing @@ Expression -- the class of all expressions
Thing ^ Expression -- the class of all expressions
Thing ^** Expression -- the class of all expressions
Thing ^^ Expression -- the class of all expressions
Thing _ Expression -- the class of all expressions
Thing | Expression -- the class of all expressions
Thing |- Expression -- the class of all expressions
Thing || Expression -- the class of all expressions
Thing and Expression -- the class of all expressions
Thing Expression -- the class of all expressions
Thing or Expression -- the class of all expressions
Thing xor Expression -- the class of all expressions
threadVariable -- create a symbol whose value in one thread is not shared with others
throw -- throw an exception
Time -- the class of all timing results
time -- time a computation
times -- multiplication
timing -- time a computation
to -- loop control
toAbsolutePath -- the absolute path version of a file name
toCC -- convert to high-precision complex number
toCC(CC) -- convert to high-precision complex number
toCC(QQ) -- convert to high-precision complex number
toCC(RR) -- convert to high-precision complex number
toCC(RR,RR) -- convert to high-precision complex number
toCC(ZZ) -- convert to high-precision complex number
toCC(ZZ,CC) -- convert to high-precision complex number
toCC(ZZ,QQ) -- convert to high-precision complex number
toCC(ZZ,QQ,QQ) -- convert to high-precision complex number
toCC(ZZ,QQ,RR) -- convert to high-precision complex number
toCC(ZZ,QQ,ZZ) -- convert to high-precision complex number
toCC(ZZ,RR) -- convert to high-precision complex number
toCC(ZZ,RR,QQ) -- convert to high-precision complex number
toCC(ZZ,RR,RR) -- convert to high-precision complex number
toCC(ZZ,RR,ZZ) -- convert to high-precision complex number
toCC(ZZ,ZZ) -- convert to high-precision complex number
toCC(ZZ,ZZ,QQ) -- convert to high-precision complex number
toCC(ZZ,ZZ,RR) -- convert to high-precision complex number
toCC(ZZ,ZZ,ZZ) -- convert to high-precision complex number
toExternalString -- convert to a readable string
toField -- declare that a ring is a field
toField(Ring) -- declare that a ring is a field
toList -- create a list
toList(BasicList) -- create a list
toList(Set) -- create a list
toList(String) -- create a list
toList(Thing) -- create a list
toLower -- convert to lower case
top level loop
topCoefficients -- first variable and its coefficient of a polynomial or matrix
topCoefficients(Matrix) -- first variable and its coefficient of a polynomial or matrix
topCoefficients(RingElement) -- first variable and its coefficient of a polynomial or matrix
topcom
topLevelMode -- the current top level mode
Tor -- Tor module
Tor_ZZ(Ideal,Ideal) -- compute a Tor module
Tor_ZZ(Ideal,Module) -- compute a Tor module
Tor_ZZ(Ideal,Ring) -- compute a Tor module
Tor_ZZ(Module,Ideal) -- compute a Tor module
Tor_ZZ(Module,Module) -- compute a Tor module
Tor_ZZ(Module,Ring) -- compute a Tor module
Toric -- compute a Gröbner basis
toRR -- convert to high-precision real number
toRRi -- construct an interval
Torsion -- value of an optional argument
toSequence -- convert to sequence
toSequence(BasicList) -- convert to sequence
toSequence(String) -- convert to sequence
toSequence(Thing) -- convert to sequence
toString -- convert to a string
TotalPairs -- whether to display the total number of S-pairs
toUpper -- convert to upper case
trace -- trace of a matrix
trace(Matrix) -- trace of a matrix
transpose -- transpose a table or a matrix
transpose(ChainComplexMap) -- transpose a map of chain complexes
transpose(List) -- transposes a table
transpose(Matrix) -- transpose a matrix
transpose(MutableMatrix) -- transpose a matrix
trim -- minimize generators and relations
trim(...,Strategy=>...) -- a Strategy option value
trim(Ideal) -- minimize generators and relations
trim(Module) -- minimize generators and relations
trim(MonomialIdeal) -- minimize generators and relations
trim(QuotientRing) -- minimize generators and relations
trim(Ring) -- minimize generators and relations
true
Truncate -- an optional argument
truncate
truncate(Constant) -- round a number toward zero
truncate(Number) -- round a number toward zero
truncateOutput
truncateOutput(InfiniteNumber)
truncateOutput(ZZ)
try -- catch an error
tutorial -- convert documentation from tutorial format
Tutorial: Canonical Embeddings of Plane Curves and Gonality
Tutorial: Divisors
Tutorial: Elementary uses of Groebner bases
Tutorial: Fano varieties
two dimensional formatting
Type -- the class of all types
TypicalValue -- make a new method function
typicalValues -- types of values returned by functions
ultimate -- ultimate value for an iteration
ultimate(Function,Thing) -- ultimate value for an iteration
unbag
unbag(Bag)
uncurry -- uncurry a function
Undo -- an optional argument
undocumented -- declare that something need not be documented
undocumented(List) -- declare that something need not be documented
undocumented(Thing) -- declare that something need not be documented
uniform -- whether all elements of a list are the same class
uninstallAllPackages
uninstallPackage -- uninstall a package and remove its documentation
uninstallPackage(...,InstallPrefix=>...) -- uninstall a package and remove its documentation
uninstallPackage(Package) -- uninstall a package and remove its documentation
uninstallPackage(String) -- uninstall a package and remove its documentation
Unique -- do not return repeated polynomial roots
unique -- eliminate duplicates from a list
unique(Sequence) -- eliminate duplicates from a list
unique(VisibleList) -- eliminate duplicates from a list
uniquePermutations -- produce all unique permutations of a list
uniquePermutations(VisibleList) -- produce all unique permutations of a list
uniquePermutations(ZZ) -- produce all unique permutations of a list
Unmixed -- value of an optional argument
unsequence -- extract the single element from a sequence of length 1
unstack -- list the rows of a net
unstack(Net) -- list the rows of a net
Up -- monomial ordering
UpdateOnly -- only copies of newer files should replace files
UpperTriangular -- an optional argument
URL -- a type representing a URL
Usage -- ways to use a function in a documentation node
use -- install or activate object
use(Monoid) -- install ring variables and ring operations
use(Package) -- activate a previously dismissed package
use(Ring) -- install ring variables and ring operations
UseCachedExampleOutput -- an optional argument
UseHilbertFunction -- an optional argument
UserMode -- an optional argument
userSymbols -- a list of the user's symbols
UseSyzygies -- compute a Gröbner basis
using functions
using functions with optional inputs
using hooks
using Macaulay2 with emacs
using Macaulay2 with emacs after it has been set up
using Macaulay2 with TeXmacs
using packages -- Macaulay2 packages
using sockets
utf8 -- encode and decode unicode utf-8-encoded strings
utf8check -- encode and decode unicode utf-8-encoded strings
utf8substring -- extract part of a utf8 string
valid names
value -- evaluate
value(Expression) -- the class of all expressions
value(IndexedVariable) -- retrieve the value of an indexed variable
value(Pseudocode) -- execute pseudocode
value(String) -- evaluate a string
value(Symbol) -- retrieve the value of a symbol
values -- values in a hash table
values(HashTable) -- values in a hash table
Variable -- specify a name for a variable
VariableBaseName -- an optional argument
Variables -- an optional argument
variables
varieties
Variety -- the class of all algebraic varieties
variety -- get the variety
variety(CoherentSheaf)
variety(Ideal) -- the closed projective subvariety defined by an ideal
variety(Ring) -- the variety previously associated to a given ring
variety(RingElement) -- the variety previously associated to a given ring
variety(SheafOfRings) -- get the variety
vars -- variables
vars(List) -- a sequence of variables
vars(Monoid) -- list of variables in a monoid
vars(Ring) -- row matrix of the variables
vars(Sequence) -- a sequence of variables
vars(ZZ) -- a sequence of variables
Vector -- the class of all elements of free modules that are handled by the engine
vector -- make a vector
Vector ** Vector -- tensor product
Vector + Vector -- a unary or binary operator, usually used for addition
Vector - Vector -- a unary or binary operator, usually used for negation or subtraction
Vector == Vector -- equality
Vector || Vector -- join Vectors
vector(List) -- make a vector
vector(Matrix) -- make a vector
VectorExpression -- the class of all vector expressions
Verbose -- request verbose feedback
Verbosity -- an optional argument
Verify -- an optional argument
Version -- an optional argument
version -- information about this version of the program
VerticalList -- a type of visible self-initializing list that prints vertically
VerticalSpace -- an optional argument
viewHelp -- view online documentation in a web browser
viewHelp(DocumentTag) -- view online documentation in a web browser
viewHelp(String) -- view online documentation in a web browser
viewHelp(Thing) -- view online documentation in a web browser
viewHelp(ZZ) -- search the documentation
viewing the symbols defined so far
VirtualTally
VirtualTally ** VirtualTally -- Cartesian product of tallies
VirtualTally + VirtualTally -- union of tallies
VirtualTally - VirtualTally
VirtualTally / Command -- apply a function to elements of a list
VirtualTally / Function -- apply a function to elements of a list
VirtualTally ? VirtualTally -- comparison operator
VirtualTally ^** ZZ -- Cartesian power of sets and tallies
VirtualTally _ Thing -- get a count from a tally
VisibleList -- the class of all visible lists
VisibleList / Command -- apply a function to elements of a list
VisibleList / Function -- apply a function to elements of a list
VisibleList / SelfInitializingType -- apply a function to elements of a list
VisibleList == VisibleList -- equality
VisibleList _ List -- get some entries of a list
VisibleList _ ZZ -- get element from list
w3
wait -- wait for child process
WebApp -- the web app top level printing method
wedgeProduct -- the exterior multiplication map
wedgeProduct(ZZ,ZZ,Module) -- the exterior multiplication map
weightRange -- the pair of lowest and highest weights of the monomials
weightRange(List,RingElement) -- the pair of lowest and highest weights of the monomials
weightRange(RingElement) -- the pair of lowest and highest weights of the monomials
Weights -- assigning weights to the variables
Weyl algebras
WeylAlgebra -- an optional argument
what a class is
what is a Groebner basis?
when -- a keyword
while -- while loops
width -- width of a file or net
width(File) -- get window width
width(Net) -- width of a net
width(String) -- width of a net
working with multiple rings
Wrap -- a key for methods for wrapping printed output
wrap
WrapperType -- a class of lists with abbreviated constructors
WrapperType List -- a class of lists with abbreviated constructors
WrapperType Sequence -- a class of lists with abbreviated constructors
WrapperType Thing -- a class of lists with abbreviated constructors
writing documentation
xor -- exclusive disjunction
youngest -- the youngest member of a sequence
zero -- whether something is zero
ZeroExpression -- the class of all zero expressions
zeta -- Riemann zeta function
zeta(RR) -- Riemann zeta function
ZZ -- the class of all integers
ZZ ! -- factorial
ZZ % MonomialIdeal -- a binary operator, usually used for remainder and reduction
ZZ % ZZ -- a binary operator, usually used for remainder and reduction
ZZ & ZZ -- logical and
ZZ * BettiTally -- the class of all Betti tallies
ZZ * CC -- a binary operator, usually used for multiplication
ZZ * ProjectiveHilbertPolynomial -- a binary operator, usually used for multiplication
ZZ * QQ -- a binary operator, usually used for multiplication
ZZ * RR -- a binary operator, usually used for multiplication
ZZ * RRi -- a binary operator, usually used for multiplication
ZZ * ZZ -- a binary operator, usually used for multiplication
ZZ + CC -- a unary or binary operator, usually used for addition
ZZ + ChainComplexMap -- a unary or binary operator, usually used for addition
ZZ + ProjectiveHilbertPolynomial -- a unary or binary operator, usually used for addition
ZZ + QQ -- a unary or binary operator, usually used for addition
ZZ + RR -- a unary or binary operator, usually used for addition
ZZ + RRi -- a unary or binary operator, usually used for addition
ZZ + ZZ -- a unary or binary operator, usually used for addition
ZZ - CC -- a unary or binary operator, usually used for negation or subtraction
ZZ - ProjectiveHilbertPolynomial -- a unary or binary operator, usually used for negation or subtraction
ZZ - QQ -- a unary or binary operator, usually used for negation or subtraction
ZZ - RR -- a unary or binary operator, usually used for negation or subtraction
ZZ - RRi -- a unary or binary operator, usually used for negation or subtraction
ZZ - ZZ -- a unary or binary operator, usually used for negation or subtraction
ZZ .. ZZ -- sequences of consecutive integers
ZZ ..< ZZ -- sequences of consecutive integers
ZZ / CC -- a binary operator, usually used for division
ZZ / QQ -- a binary operator, usually used for division
ZZ / RR -- a binary operator, usually used for division
ZZ / RRi -- a binary operator, usually used for division
ZZ / ZZ -- a binary operator, usually used for division
ZZ // MonomialIdeal -- a binary operator, usually used for quotient
ZZ // ZZ -- a binary operator, usually used for quotient
ZZ : Thing -- repeat an item
ZZ << ZZ
ZZ == CC -- equality
ZZ == ChainComplex -- equality
ZZ == ChainComplexMap -- equality
ZZ == GradedModuleMap -- equality
ZZ == Ideal -- equality
ZZ == Module -- equality
ZZ == MonomialIdeal -- equality
ZZ == MutableMatrix -- equality
ZZ == ProjectiveHilbertPolynomial -- equality
ZZ == QQ -- equality
ZZ == Ring -- equality
ZZ == RingElement -- equality
ZZ == RingMap -- equality
ZZ == RR -- equality
ZZ == RRi -- equality
ZZ == ZZ -- equality
ZZ >> ZZ
ZZ ^^ ZZ -- logical exclusive-or
ZZ _ Module -- integers or zero element
ZZ | Matrix -- join matrices horizontally
ZZ | String -- join strings or nets
ZZ | ZZ -- logical or
ZZ || Matrix -- join matrices vertically
ZZ ~ -- logical not