Function -- the class of all functions

Types of function :

• CompiledFunction -- the class of all compiled functions
• CompiledFunctionClosure -- the class of all compiled function closures
• FunctionClosure -- the class of all function closures

Functions and methods returning a function :

• Function @@ Function -- composition of functions
• Function _ Thing -- attach the first argument to a function of two or more arguments
• "Boolean >> Function" -- see OptionTable >> Function -- attaching options to a function
• "List >> Function" -- see OptionTable >> Function -- attaching options to a function
• OptionTable >> Function -- attaching options to a function
• "profile(Function)" -- see profile -- profile a function

Methods that use a function :

• "? Function" -- see ? Symbol -- view brief documentation of a symbol
• "about(Function)" -- see about -- search the documentation
• "accumulate(Function,Thing,VisibleList)" -- see accumulate -- apply a binary operator repeatedly
• "accumulate(Function,VisibleList)" -- see accumulate -- apply a binary operator repeatedly
• "accumulate(VisibleList,Function)" -- see accumulate -- apply a binary operator repeatedly
• "accumulate(VisibleList,Thing,Function)" -- see accumulate -- apply a binary operator repeatedly
• "addHook(MutableHashTable,Thing,Function)" -- see addHook -- add a hook function to an object for later processing
• "addHook(Sequence,Function)" -- see addHook -- add a hook function to an object for later processing
• "addHook(Symbol,Function)" -- see addHook -- add a hook function to an object for later processing
• 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(ZZ,Function)" -- see 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
• 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(ZZ,Function)" -- see 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
• 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(ZZ,Function) -- apply a function to {0,..., n-1}
• "applyKeys(HashTable,Function)" -- see applyKeys -- apply a function to each key in a hash table
• "applyKeys(HashTable,Function,Function)" -- see applyKeys -- apply a function to each key in a hash table
• "applyPairs(HashTable,Function)" -- see applyPairs -- apply a function to each pair in a hash table
• "applyValues(HashTable,Function)" -- see applyValues -- apply a function to each value in a hash table
• "code(Function)" -- see code -- display source code
• "createTask(Function)" -- see createTask -- create a task
• "createTask(Function,Thing)" -- see createTask -- create a task
• "fold(Function,Thing,VisibleList)" -- see fold -- apply a binary operator repeatedly
• "fold(Function,VisibleList)" -- see fold -- apply a binary operator repeatedly
• "fold(VisibleList,Function)" -- see fold -- apply a binary operator repeatedly
• "fold(VisibleList,Thing,Function)" -- see fold -- apply a binary operator repeatedly
• "frames(Function)" -- see frames -- get the frames associated to a closure
• Function \ Set (missing documentation)
• Function Thing -- function application
• /// Function \ Ideal /// -- see Ideal / Function -- apply a function to generators of an ideal
• Ideal / Function -- apply a function to generators of an ideal
• "installAssignmentMethod(Symbol,HashTable,Function)" -- see installAssignmentMethod -- install methods assignment to the value of an operator
• "installAssignmentMethod(Symbol,HashTable,HashTable,Function)" -- see installAssignmentMethod -- install methods assignment to the value of an operator
• "localDictionaries(Function)" -- see localDictionaries -- get local dictionaries
• "locate(Function)" -- see locate -- locate source code
• "new Manipulator from Function" -- see Manipulator -- the class of all file manipulators
• map(ChainComplex,ChainComplex,Function) -- make a map of chain complexes
• "map(GradedModule,GradedModule,Function)" -- see map(ChainComplex,ChainComplex,Function) -- make a map of chain complexes
• map(Module,Module,Function) -- create a matrix by specifying a function that gives each entry
• map(Module,ZZ,Function) -- create a matrix from a free module by specifying a function that gives each entry
• "memoize(Function)" -- see memoize -- record results of function evaluation for future use
• "memoize(Function,List)" -- see memoize -- record results of function evaluation for future use
• "merge(HashTable,HashTable,Function)" -- see merge -- merge two hash tables
• "mergePairs(BasicList,BasicList,Function)" -- see mergePairs -- merge sorted lists of pairs
• methodOptions(Function) -- recover the options used when a method function was created
• options(Function) -- get optional arguments and default values for a function that accepts them
• "package(Function)" -- see package -- get containing package
• "partition(Function,VirtualTally)" -- see partition -- partition a set or list by values of a function
• "partition(Function,VirtualTally,VisibleList)" -- see partition -- partition a set or list by values of a function
• "partition(Function,VisibleList)" -- see partition -- partition a set or list by values of a function
• "partition(Function,VisibleList,VisibleList)" -- see partition -- partition a set or list by values of a function
• "position(VisibleList,Function)" -- see position -- the first element of a list satisfying a condition
• "position(VisibleList,VisibleList,Function)" -- see position -- the first element of a list satisfying a condition
• "positions(MutableList,Function)" -- see positions -- which elements of a list satisfy a condition
• "positions(VisibleList,Function)" -- see positions -- which elements of a list satisfy a condition
• 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
• "profile(String,Function)" -- see profile -- profile a function
• "scan(BasicList,Function)" -- see scan -- apply a function to each element in a list or sequence
• "scan(ZZ,Function)" -- see scan -- apply a function to each element in a list or sequence
• "scanKeys(Database,Function)" -- see scanKeys -- apply a function to each key in a hash table or database
• "scanKeys(HashTable,Function)" -- see scanKeys -- apply a function to each key in a hash table or database
• "scanLines(Function,List)" -- see scanLines -- apply a function to each line of a file
• "scanLines(Function,String)" -- see scanLines -- apply a function to each line of a file
• "scanPairs(HashTable,Function)" -- see scanPairs -- apply a function to the pairs in a hash table
• "scanValues(HashTable,Function)" -- see scanValues -- apply a function to each value in a hash table or database
• "schedule(Function)" -- see schedule -- schedule a task for execution
• "schedule(Function,Thing)" -- see schedule -- schedule a task for execution
• select(BasicList,Function) -- select elements from a list
• select(HashTable,Function) -- select part of a hash table
• 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
• Set / Function (missing documentation)
• sort(List,Function) (missing documentation)
• "sublists(VisibleList,Function)" -- see sublists -- process interspersed subsequences of a visible list
• "sublists(VisibleList,Function,Function)" -- see sublists -- process interspersed subsequences of a visible list
• "sublists(VisibleList,Function,Function,Function)" -- see sublists -- process interspersed subsequences of a visible list
• "sublists(VisibleList,Function,Function,Nothing)" -- see sublists -- process interspersed subsequences of a visible list
• "sublists(VisibleList,Function,Nothing)" -- see sublists -- process interspersed subsequences of a visible list
• "sublists(VisibleList,Function,Nothing,Function)" -- see sublists -- process interspersed subsequences of a visible list
• "sublists(VisibleList,Function,Nothing,Nothing)" -- see sublists -- process interspersed subsequences of a visible list
• 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
• /// Function \\ Thing /// -- see Thing // Function -- apply a function
• Thing // Function -- apply a function
• "ultimate(Function,Thing)" -- see ultimate -- ultimate value for an iteration
• /// Function \ VirtualTally /// -- see VisibleList / Function -- apply a function to elements of a list
• /// Function \ VisibleList /// -- see VisibleList / Function -- apply a function to elements of a list
• "List / Function" -- see VisibleList / Function -- apply a function to elements of a list
• "VirtualTally / Function" -- see VisibleList / Function -- apply a function to elements of a list
• VisibleList / Function -- apply a function to elements of a list

Fixed objects of class Function :

• about -- search the documentation
• abs -- absolute value function
• accumulate -- apply a binary operator repeatedly
• acos -- arccosine
• acosh -- inverse hyperbolic cosine
• acot -- compute the inverse cotangent
• "addCancelTask" -- see addCancelTask(Task,Task) -- specify that the completion of one task triggers the cancellation of another
• "addDependencyTask" -- see 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
• addStartFunction -- add a startup function
• "addStartTask" -- see addStartTask(Task,Task) -- schedule a task upon completion of another
• "adjoint'" -- see adjoint'(Matrix,Module,Module) -- an adjoint map
• "adjoint" -- see adjoint(Matrix,Module,Module) -- an adjoint map
• agm -- arithmetic-geometric mean
• alarm -- set an alarm
• all -- whether all elements satisfy a specified condition
• ambient -- ambient free module of a subquotient, or ambient ring
• analyticSpread -- Compute the analytic spread of a module or ideal
• "ancestor" -- see ancestor(Type,Type) -- whether one type is an ancestor of another
• ancestors -- the ancestors of something
• annihilator (missing documentation) -- the annihilator ideal
• antipode -- antipode for skew commuting polynomial rings
• any -- whether any elements satisfy a specified condition
• append -- add an element to the end of a list
• applicationDirectory -- the path to the user's application directory
• apply -- apply a function to each element
• applyKeys -- apply a function to each key in a hash table
• applyPairs -- 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
• apropos -- symbols matching a pattern
• arXiv -- link to an arXiv article
• ascii -- ASCII character conversion
• asin -- arcsine
• asinh -- inverse hyperbolic sine
• assert -- assert something is true
• associatedPrimes -- find associated primes
• atan -- compute the arctangent
• "atan2" -- see atan2(RR,RR) -- compute an angle of a certain triangle
• "atEndOfFile" -- see atEndOfFile(File) -- test for end of file
• autoload -- arrange for a function to be loaded automatically
• baseFilename -- the base part of a filename or path
• baseName -- the base name of a generator
• baseRing -- produce the ring from which a ring was formed
• basis -- basis or generating set of all or part of a ring, ideal or module
• beginDocumentation -- start documentation section of a package
• benchmark -- accurate timing of execution
• BesselJ -- Bessel function of the first kind
• BesselY -- Bessel function of the second kind
• betti -- display degrees
• between -- insert something between elements of a list
• binomial -- binomial coefficient
• "borel" -- see borel(Matrix) -- make a Borel fixed submodule
• cacheValue -- cache values of functions in their arguments
• "cancelTask" -- see cancelTask(Task) -- stop a task
• capture -- evaluate Macaulay2 code and capture the output
• ceiling -- ceiling function
• centerString -- center a string or net
• chainComplex -- make a chain complex
• char -- computes the characteristic of the ring or field
• characters -- get characters from a string
• check -- perform tests of a package
• chi
• class -- class of an object
• clean -- Set to zero elements that are approximately zero
• clearEcho -- turn off echoing
• code -- display source code
• codim -- compute the codimension
• coefficient -- coefficient of a monomial
• coefficientRing -- get the coefficient ring
• coefficients -- monomials and their coefficients
• cohomology -- general cohomology functor
• coimage -- coimage of a map
• cokernel -- 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
• columnate -- arrange strings in columns
• columnMult -- multiply a column by a ring element
• columnPermute -- permute some columns
• columnRankProfile (missing documentation)
• columnSwap -- interchange columns
• combine -- combine hash tables
• commandInterpreter -- the top level command interpreter
• commonest -- the most common elements of a list or tally
• commonRing -- find smallest containing ring
• comodule -- submodule to quotient module
• "complement" -- see complement(Matrix) -- find the minimal generators for cokernel of a matrix (low level form)
• complete
• components -- list the components of a direct sum
• "compose" -- see compose(Module,Module,Module) -- composition as a pairing on Hom-modules
• compositions -- list the compositions of an integer
• compress -- extract nonzero columns from a matrix
• concatenate -- join strings
• conductor -- the conductor of a finite ring map
• cone -- mapping cone of a chain map
• conjugate -- complex conjugate
• connectionCount -- the number of connections
• content -- the content of a polynomial
• contract -- contract one matrix by another
• "contract'" -- see contract'(Matrix,Matrix) -- contract a matrix by a matrix, the dual notion
• conwayPolynomial -- provide a Conway polynomial
• copy -- copy an object
• "copyDirectory" -- see copyDirectory(String,String)
• "copyFile" -- see copyFile(String,String)
• cos -- compute the cosine
• cosh -- compute the hyperbolic cosine
• cot -- cotangent
• cotangentSheaf -- cotangent sheaf of a projective variety
• coth -- hyperbolic cotangent
• cover -- get the covering free module
• "coverMap" -- see 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
• csc -- cosecant
• csch -- hyperbolic cosecant
• currentDirectory -- current working directory
• currentLineNumber -- current line number of the current input file
• currentTime -- get the current time
• debug -- open the private dictionary of a package
• decompose
• deepSplice -- remove all levels of subsequences from a sequence or list
• default -- default member of a family
• degree
• degreeLength -- the number of degrees
• degrees -- degrees of generators
• degreesMonoid -- get the monoid of degrees
• degreesRing -- the ring of degrees
• delete -- delete some elements of a list
• demark -- insert a string between elements of a list of strings
• denominator -- denominator of a fraction
• depth -- depth of a net
• describe -- real description
• determinant -- determinant of a matrix
• diagonalMatrix -- make a diagonal matrix
• dictionary -- determine the dictionary to which a symbol belongs
• diff -- differentiate or take difference
• "diff'" -- see diff'(Matrix,Matrix) -- differentiate a matrix by a matrix, the dual notion
• difference -- difference
• dim -- compute the Krull dimension
• directSum -- direct sum of modules or maps
• disassemble -- disassemble pseudocode or a function
• "discriminant" -- see discriminant(RingElement,RingElement)
• dismiss -- dismiss a package
• distinguished -- Compute the distinguished subvarieties of a pullback, intersection or cone
• divideByVariable -- divide all columns by a (power of a) variable
• doc -- a simple documentation function
• "multidoc" -- see doc -- a simple documentation function
• document -- create a documentation node
• "makeDocumentTag" -- see DocumentTag -- the class of all document tags
• drop -- drop some elements from a list or sequence.
• dual -- dual module or map
• "eagonNorthcott" -- see eagonNorthcott(Matrix) -- Eagon-Northcott complex of a matrix of linear forms
• echoOff -- turn off echoing
• echoOn -- turn on echoing
• eigenvalues -- find eigenvalues of a matrix
• eigenvectors -- find eigenvectors of a matrix over RR or CC
• eint -- exponential integral
• elements -- list of elements
• eliminate
• End -- module of endomorphisms
• endPackage -- end a new package
• entries -- lists the entries of a matrix
• erase -- remove a global symbol
• erf -- error function
• erfc -- complementary error function
• error -- deliver error message
• euler -- Euler characteristic
• eulers -- list the sectional Euler characteristics
• even -- tell whether an integer is even
• EXAMPLE -- hypertext EXAMPLE item
• examples -- list the examples in documentation
• exec -- execute another program
• exp -- exponential function
• expectedReesIdeal -- symmetric algebra ideal plus jacobian dual
• expm1 -- exponential minus 1
• exponents -- the exponents of a polynomial
• export -- export symbols from a package
• exportFrom -- export symbols from a package's private dictionary
• exportMutable -- export mutable symbols from a package
• expression -- convert to an expression
• extend -- extend a module map to a chain map, if possible
• exteriorPower -- exterior power
• factor -- factor a ring element or a ZZ-module
• Fano -- Fano scheme
• fileExecutable
• fileExists -- whether a file exists
• fileLength -- the length of a file
• fileMode -- set or get file mode
• fileReadable
• fileTime -- get or set file modified time
• fileWritable
• fillMatrix -- fill a mutable matrix with random numbers
• findFiles -- find files recursively
• findHeft -- find a heft vector for a list of degrees
• findProgram -- load external program
• "findSynonyms" -- see findSynonyms(Symbol) -- find synonyms of symbols
• first -- first element of a list
• firstkey -- get the first key
• fittingIdeal -- Fitting ideal of a module
• flagLookup -- flag a symbol
• flatten -- flatten a nested list or a matrix
• flattenRing -- write a ring as a (quotient of a) polynomial ring
• "flip" -- see flip(Module,Module) -- matrix of commutativity of tensor product
• floor -- floor function
• fold -- apply a binary operator repeatedly
• forceGB -- declare that the columns of a matrix are a Gröbner basis
• fork -- fork the process
• format -- format a string or a real number
• formation -- recover the methods used to make a module
• frac -- construct a fraction field
• fraction
• frames -- get the frames associated to a closure
• fromDividedPowers -- Translates from divided power monomial basis to ordinary monomial basis
• fromDual -- Ideal from inverse system
• functionBody -- get the body of a function
• Gamma -- Gamma function
• gb -- compute a Gröbner basis
• gbRemove -- remove Gröbner basis
• gbSnapshot -- the Gröbner basis matrix as so far computed
• gcd -- greatest common divisor
• gcdCoefficients -- gcd with coefficients
• gcdLLL -- compute the gcd of integers, and small multipliers
• GCstats (missing documentation)
• genera -- list of the successive linear sectional arithmetic genera
• generateAssertions -- generate assert statements from experimental input
• generator -- provide a single generator
• generators -- provide matrix or list of generators
• genericMatrix -- make a generic matrix of variables
• genericSkewMatrix -- make a generic skew symmetric matrix of variables
• genericSymmetricMatrix -- make a generic symmetric matrix
• genus -- arithmetic genus
• get -- get the contents of a file
• getc -- get a byte
• getChangeMatrix -- get the change of basis matrix
• getenv -- 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
• getSymbol -- make a global user symbol from a string
• getWWW -- get a web page
• GF -- make a finite field
• globalAssign -- global assignment, in function form
• globalAssignFunction -- the standard method for the global assignment hook
• globalAssignment -- install standard global assignment method
• globalReleaseFunction -- the standard method for the global variable release hook
• gradedModule -- make a graded module
• gradedModuleMap -- make a map of graded modules
• gramm (missing documentation)
• "graphIdeal" -- see graphIdeal(RingMap) -- the ideal of the graph of the regular map corresponding to a ring map
• "graphRing" -- see graphRing(RingMap) -- the coordinate ring of the graph of the regular map corresponding to a ring map
• "Grassmannian" -- see Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space
• groebnerBasis -- Gröbner basis, as a matrix
• groupID -- the process group identifier
• hash -- hash code of an object
• hashTable -- make a hash table
• heft -- heft vector of ring, module, graded module, or resolution
• height -- height of a net
• hermite (missing documentation)
• hilbertFunction -- the Hilbert function
• hilbertPolynomial -- compute the Hilbert polynomial
• hilbertSeries -- compute the Hilbert series
• hold -- hold something in a holder expression
• Hom -- module of homomorphisms
• homogenize -- homogenize with respect to a variable
• homology -- general homology functor
• homomorphism -- get the homomorphism from element of Hom
• homomorphism' -- get the element of Hom from a homomorphism
• hooks -- list hooks attached to a key
• horizontalJoin -- join nets or strings horizontally
• html -- convert to html format
• "httpHeaders" -- see httpHeaders(String) -- prepend http headers to a string
• hypertext -- prepare hypertext for display
• icFracP -- compute the integral closure in prime characteristic
• icFractions -- fractions integral over an affine domain
• icMap -- natural map from an affine domain into its integral closure
• icPIdeal -- compute the integral closure in prime characteristic of a principal ideal
• ideal -- make an ideal
• idealizer -- compute Hom(I,I) as a quotient ring
• identity -- the identity function
• image -- image of a map
• imaginaryPart -- imaginary part
• importFrom -- import symbols to the current private dictionary
• independentSets -- some size-maximal independent subsets of variables modulo an ideal
• index -- numeric index of a ring variable
• indices -- indices of a polynomial; also components for a direct sum
• inducedMap -- compute an induced map
• inducesWellDefinedMap -- whether a map is well defined
• info -- convert hypertext to info format
• infoHelp -- view documentation in Info format
• input -- read Macaulay2 commands and echo
• insert -- copy a list, inserting an element
• installAssignmentMethod -- install methods assignment to the value of an operator
• installedPackages
• installHilbertFunction -- install a Hilbert function without computation
• installMethod -- install methods
• installPackage -- load and install a package and its documentation
• instance -- whether something has a certain type
• "instances" -- see instances(Type)
• integralClosure -- integral closure of an ideal or a domain
• integrate -- numerical integration
• intersect -- compute an intersection
• intersectInP -- Compute distinguished varieties for an intersection in A^n or P^n
• inverse -- compute the inverse
• inversePermutation -- inverse permutation
• inverseSystem -- Inverse systems with equivariance
• irreducibleCharacteristicSeries -- irreducible characteristic series of an ideal
• irreducibleDecomposition -- express a monomial ideal as an intersection of irreducible monomial ideals
• isAffineRing -- whether something is an affine ring
• isANumber -- whether a number is not not-a-number
• isBorel -- whether an ideal is fixed by upper triangular changes of coordinates
• "isCanceled" -- see isCanceled(Task) -- whether a task has been canceled
• isCommutative -- whether a ring is commutative
• isConstant -- whether a ring element is constant
• isDirectory -- whether a file is a directory
• isDirectSum -- whether something is a direct sum
• isField -- whether something is a field
• isFinite -- whether a number is finite
• isFinitePrimeField -- whether a ring is a finite prime field
• isFreeModule -- whether something is a free module
• isGlobalSymbol -- whether a global symbol with a given name exists
• isHomogeneous -- whether something is homogeneous (graded)
• isIdeal -- whether something is an ideal
• isInfinite -- whether a number is infinite
• isInjective -- whether a map is injective
• "isInputFile" -- see isInputFile(File) -- whether a file is open for input
• isIsomorphism -- whether a map is an isomorphism
• isLinearType -- Determine whether module has linear type
• "isListener" -- see isListener(File) -- whether a file is open for listening
• isLLL -- is a basis an LLL basis?
• isModule -- whether something is a module
• isMonomialIdeal -- whether something is a monomial ideal
• isNormal -- determine if a reduced ring is normal
• isOpen -- whether a file or database is open
• "isOutputFile" -- see isOutputFile(File) -- whether a file is open for output
• isPolynomialRing -- whether someting is a polynomial ring
• isPrimary -- determine whether a submodule is primary
• isPrime -- whether a integer or polynomial is prime
• isPrimitive -- whether an element is a primitive element of a finite field
• "isPseudoprime" -- see isPseudoprime(ZZ) -- whether an integer is probably prime
• isQuotientModule -- whether something is evidently a quotient of a free module
• isQuotientOf -- whether one thing is a quotient of another
• isQuotientRing -- whether something is a quotient ring
• "isReady" -- see isReady(File) -- whether a file has data available for reading
• isReal -- whether a number is real
• isReduction -- Determine whether an ideal is a reduction
• isRegularFile -- whether a file is a regular file
• isRing -- whether something is a ring
• isSkewCommutative -- whether a ring has skew commuting variables
• isSorted -- whether a list is sorted
• isSquareFree -- whether something is square free monomial ideal
• isStandardGradedPolynomialRing -- Checks whether a ring is a polynomial ring over a field with variables of degree 1
• isSubmodule -- whether a module is evidently a submodule of a free module
• "isSubquotient" -- see isSubquotient(Module,Module) -- check whether a module is a subquotient of another
• isSubset -- whether one object is a subset of another
• isSupportedInZeroLocus -- whether support of a module is contained in the zero locus of the (irrelevant) ideal
• isSurjective -- whether a map is surjective
• isTable -- whether something is a list of lists of equal length
• isUnit -- whether a ring element is a unit
• isWellDefined -- whether a map is well defined
• isWeylAlgebra (missing documentation)
• jacobian -- the Jacobian matrix of partial derivatives
• jacobianDual -- Computes the 'jacobian dual', part of a method of finding generators for Rees Algebra ideals
• join -- join lists and sequences
• kernel -- kernel of a ringmap, matrix, or chain complex
• kernelLLL (missing documentation)
• kernelOfLocalization -- the kernel of the localization map
• keys -- keys used in a hash table, dictionary, or database
• kill -- kill a process
• koszul -- Koszul complex or specific matrix in the Koszul complex
• last -- last element of a list
• lcm -- least common multiple
• leadCoefficient -- the coefficient of the leading term
• leadComponent -- the leading component(s) of a vector or matrix
• leadMonomial -- the leading monomial of a ring element
• leadTerm -- get the greatest term
• length -- length of an object
• lift -- lift to another ring
• liftable -- whether lifting to another ring is possible
• limitFiles
• limitProcesses
• lines -- split a string into lines
• "linkFile" -- see linkFile(String,String) -- make a new link to a file
• listForm -- convert to list form
• listSymbols -- compact display of symbols and their values
• LLL -- compute an LLL basis
• lngamma -- logarithm of the Gamma function
• load -- read Macaulay2 commands
• loadPackage -- load a package
• localDictionaries -- get local dictionaries
• localize -- localize an ideal at a prime ideal
• locate -- locate source code
• log -- logarithm function
• log1p -- logarithm of 1+x
• lookup -- look up methods
• lookupCount -- reference count for a symbol
• LUdecomposition -- LU decomposition
• "makeDirectory" -- see makeDirectory(String) -- make a directory
• makePackageIndex -- create an index of installed packages
• makeS2 -- compute the S2ification of a reduced ring
• map -- make a map
• "markedGB" -- see markedGB(Matrix,Matrix) -- make a marked Gröbner basis
• match -- regular expression matching
• mathML -- convert to MathML format
• matrix -- make a matrix
• max -- yields the maximum element in a list or sequence
• maxPosition -- position of the largest element
• member -- test membership in a list or set
• memoize -- record results of function evaluation for future use
• merge -- merge two hash tables
• mergePairs -- merge sorted lists of pairs
• method -- make a new method function
• "methodOptions" -- see methodOptions(Function) -- recover the options used when a method function was created
• methods -- list methods
• min -- yields the minimum element in a list or sequence
• mingens -- minimal generator matrix
• mingle -- mingle elements of several lists
• "minimalBetti" -- see minimalBetti(Ideal) -- minimal betti numbers of (the mininimal free resolution of) a homogeneous ideal or module
• minimalPresentation -- compute a minimal presentation
• minimalPrimes -- minimal primes of an ideal
• minimalReduction -- Find a minimal reduction of an ideal
• minimizeFilename -- minimize a file name
• "minors" -- see minors(ZZ,Matrix) -- ideal generated by minors
• minPosition -- position of the smallest element
• minus -- additive inverse
• mkdir
• mod -- reduce modulo an integer
• module -- make or get a module
• "modulo" -- see modulo(Matrix,Matrix) -- find the pre-image (pullback) of image of a map (low level version)
• monoid -- make or retrieve a monoid
• monomialCurveIdeal -- make the ideal of a monomial curve
• monomialIdeal -- make a monomial ideal
• monomials -- matrix of monomials in a ring element or matrix
• monomialSubideal -- find the largest monomial ideal in an ideal
• "moveFile" -- see moveFile(String,String)
• multidegree -- multidegree
• "multigraded" -- see multigraded(BettiTally) -- convert a Betti tally into a multigraded Betti tally
• multiplicity -- Compute the Hilbert-Samuel multiplicity of an ideal
• mutable -- whether something may be modified
• "mutableIdentity" -- see mutableIdentity(Ring,ZZ) -- make a mutable identity matrix
• mutableMatrix -- make a mutable matrix
• nanosleep -- sleep for a given number of nanoseconds
• needs -- read Macaulay2 commands if necessary
• needsPackage -- load a package if not already loaded
• net -- format for printing, as a net
• "netList" -- see netList(VisibleList) -- a table of boxes
• newClass -- set the class and parent of an object
• newCoordinateSystem -- change variables
• newNetFile -- create a new net file
• newPackage -- the preamble of a package
• newRing -- make a copy of a ring, with some features changed
• nextkey -- the next key in a database
• nextPrime -- compute the smallest prime greater than or equal to a given number
• norm
• normalCone -- The normal cone of a subscheme
• notImplemented -- print an 'not implemented' error message
• nullhomotopy -- make a null homotopy
• nullSpace (missing documentation)
• number -- count how many elements of a list satisfy a condition
• "numColumns" -- see numColumns(Matrix) -- number of columns in a matrix or mutable matrix
• numerator -- numerator of a fraction
• numeric -- convert to floating point
• numgens -- the number of generators
• "numRows" -- see numRows(Matrix) -- number of rows in a matrix or mutable matrix
• odd -- tell whether an integer is odd
• ofClass -- English phrases for types
• on -- trace a function each time it's run
• openDatabase -- open a database file
• openDatabaseOut -- open a database file for writing
• openFiles -- list the open files
• "openIn" -- see openIn(String) -- open an input file
• openInOut -- open an input output file
• "openListener" -- see openListener(String) -- open a port for listening
• "openOut" -- see openOut(String) -- open an output file
• "openOutAppend" -- see openOutAppend(String) -- open an output file for appending
• options -- get options
• override -- override default values for optional arguments
• pack -- pack elements of a list into several shorter lists
• package -- get containing package
• packageTemplate -- a template for a package
• pad -- pad a string or net with spaces
• pager -- display with paging
• pairs -- list the pairs in a hash table, dictionary, or basic list
• parent -- parent type of an object
• part -- select terms of a polynomial by degree(s) or weight(s)
• partition -- partition a set or list by values of a function
• partitions -- list the partitions of an integer
• parts -- display terms of a polynomial degree by degree
• pdim -- calculate the projective dimension
• peek -- examine contents of an object
• "peek'" -- see peek'(ZZ,Thing) -- examine contents of an object
• permanents -- ideal generated by square permanents of a matrix
• permutations -- produce all permutations of a list
• pfaffians -- ideal generated by Pfaffians
• "pivots" -- see pivots(Matrix) -- list of pivot locations of a matrix
• plus -- addition
• poincare -- assemble degrees into polynomial
• poincareN -- assemble degrees into polynomial
• polarize -- given a monomial ideal, computes the squarefree monomial ideal obtained via polarization
• "poly" -- see poly(String) -- make a polynomial using classic Macaulay syntax
• position -- the first element of a list satisfying a condition
• positions -- which elements of a list satisfy a condition
• power -- power
• powermod -- powers of integers mod N
• precision
• preimage -- preimage of an ideal under a ring map
• prepend -- add an element to the beginning of a list
• presentation -- presentation of a module or ring
• pretty -- a pretty printer
• primaryComponent -- find a primary component corresponding to an associated prime
• primaryDecomposition -- irredundant primary decomposition of an ideal
• print -- print something
• printString -- lowlevel function to print a string, net, or symbol
• processID -- the process identifier
• product
• profile -- profile a function
• "Proj" -- see Proj(Ring) -- make a projective variety
• projectiveHilbertPolynomial -- Hilbert polynomial of projective space
• promote -- promote to another ring
• protect -- protect a symbol
• prune -- prune, e.g., compute a minimal presentation
• pseudocode -- produce the pseudocode for a function
• pseudoRemainder -- compute the pseudo-remainder
• "pushForward" -- see pushForward(RingMap,Module)
• QRDecomposition -- compute a QR decomposition of a real matrix
• quotient -- quotient or division
• quotient' -- matrix quotient (opposite)
• quotientRemainder -- matrix quotient and remainder
• quotientRemainder' -- matrix quotient and remainder (opposite)
• radical -- the radical of an ideal
• radicalContainment -- whether an element is contained in the radical of an ideal
• random -- get a random element
• randomKRationalPoint -- Pick a random K rational point on the scheme X defined by I
• "randomMutableMatrix" -- see randomMutableMatrix(ZZ,ZZ,RR,ZZ) -- a random mutable matrix of integers
• rank -- compute the rank
• read -- read from a file
• readDirectory -- read the contents of a directory
• readlink -- readlink fn
• readPackage -- read the package preamble
• realPart -- real part
• realpath -- convert a filename to one passing through no symbolic links
• recursionDepth -- the current recursion depth
• reducedRowEchelonForm -- compute the reduced row echelon form of a matrix or mutable matrix over a field
• reduceHilbert -- reduce a Hilbert series expression
• reductionNumber -- Reduction number of one ideal with respect to another
• reesAlgebra -- Compute the defining ideal of the Rees Algebra
• reesIdeal -- Compute the defining ideal of the Rees Algebra
• regex -- evaluate a regular expression search
• regexQuote -- escape special characters in regular expressions
• registerFinalizer -- register a string that will be displayed when an object is garbage collected
• regSeqInIdeal -- a regular sequence contained in an ideal
• regularity -- compute the Castelnuovo-Mumford regularity
• relations -- the defining relations
• relativizeFilename -- relativize a file name
• remainder -- matrix remainder
• remainder' -- matrix quotient and remainder (opposite)
• remove -- remove an entry from a mutable hash table
• removeDirectory -- remove a directory
• removeFile -- remove a file
• removeLowestDimension -- remove components of lowest dimension
• reorganize -- reorganize a database file
• replace -- replacement in strings and lists
• "reshape" -- see reshape(Module,Module,Matrix) -- reshape a matrix
• resolution -- projective resolution
• "resultant" -- see resultant(RingElement,RingElement,RingElement)
• reverse -- reverse a list or sequence
• ring -- get the associated ring of an object
• ringFromFractions -- find presentation for f.g. ring
• "roots" -- see roots(RingElement) -- compute the roots of a polynomial
• "rotate" -- see rotate(ZZ,VisibleList) -- rotate a list
• round -- round a number
• rowAdd -- add a multiple of one row to another
• rowMult -- multiply a row by a ring element
• rowPermute -- permute some rows
• rowRankProfile (missing documentation)
• rowSwap -- interchange rows
• rsort -- sort a list or matrix in reverse order
• run -- run an external command
• runHooks -- run the hook functions stored in an object
• "runLengthEncode" -- see runLengthEncode(VisibleList) -- run length encoding
• runProgram -- run an external program
• same -- whether everything in a list is the same
• saturate (missing documentation) -- saturation of ideal or submodule
• scan -- apply a function to each element in a list or sequence
• scanKeys -- apply a function to each key in a hash table or database
• scanLines -- apply a function to each line of a file
• scanPairs -- apply a function to the pairs in a hash table
• scanValues -- apply a function to each value in a hash table or database
• schedule -- schedule a task for execution
• schreyerOrder -- create or obtain free modules with Schryer monomial orders
• "Schubert" -- see Schubert(ZZ,ZZ,VisibleList) -- find the Pluecker ideal of a Schubert variety
• "searchPath" -- see searchPath(List,String) -- search a path for a file
• sec -- secant
• sech -- hyperbolic secant
• seeParsing -- display parsing precedence table for Macaulay2 operators
• select -- select from a list, hash table, or string
• selectInSubring -- select columns in a subring
• "selectVariables" -- see selectVariables(List,PolynomialRing) -- make a subring of a polynomial ring generated by selected variables
• separate -- split a string into substrings using a regular expression
• "separateRegexp" -- see separate -- split a string into substrings using a regular expression
• sequence -- make a sequence
• serialNumber -- serial number of a dictionary, task, symbol, mutable hash table, or mutable list,
• set -- 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
• setup
• setupEmacs
• sheaf -- make a coherent sheaf
• "sheafHom" -- see sheafHom(CoherentSheaf,CoherentSheaf) -- sheaf Hom
• show -- display various TeX, hypertext, or a URL, in an external application
• showHtml -- convert hypertext to html and display with a browser
• showTex -- convert to TeX and display on screen
• simpleDocFrob -- a sample documentation node
• sin -- compute the sine
• singularLocus -- singular locus
• sinh -- compute the hyperbolic sine
• size -- the size of an object
• size2 -- number of binary digits to the left of the point
• sleep -- sleep for a while
• "smithNormalForm" -- see smithNormalForm(Matrix) -- smith normal form for a matrix over ZZ or a PID
• solve -- solve linear equation(s)
• someTerms -- select some terms of a polynomial
• sort -- sort a list or columns of a matrix
• sortColumns -- permutation giving sort order
• source -- source of a map
• "Spec" -- see Spec(Ring) -- make an affine variety
• specialFiber -- Special fiber of a blowup
• specialFiberIdeal -- Special fiber of a blowup
• splice -- remove subsequences from a sequence or list
• splitWWW -- separate an http response into header and body
• sqrt -- square root function
• stack -- join nets or string vertically
• stacksProject -- link to a Stacks Project tag
• standardForm -- convert to standard form
• standardPairs -- find the standard pairs of a monomial ideal
• stashValue -- stash values of functions in their arguments
• status -- status of a resolution computation
• sublists -- process interspersed subsequences of a visible list
• submatrix
• submatrix' -- exclude rows and/or columns of a matrix
• submatrixByDegrees -- submatrix consisting of rows and columns in an interval or box of degrees
• subquotient -- make a subquotient module
• subsets -- produce the subsets of a set or list
• substitute -- substituting values for variables
• substring -- extract part of a string
• subtable -- extract a subtable from a table
• sum -- compute the sum
• super -- get the ambient module
• support -- list of variables occurring in a polynomial or matrix
• SVD -- singular value decomposition of a matrix
• switch -- copy a list, switching two elements
• "sylvesterMatrix" -- see sylvesterMatrix(RingElement,RingElement,RingElement)
• symbolBody (missing documentation)
• "symlinkDirectory" -- see symlinkDirectory(String,String) -- make symbolic links for all files in a directory tree
• symlinkFile -- make a symbolic link to a file
• symmetricAlgebra -- the symmetric algebra of a module
• symmetricAlgebraIdeal -- Ideal of the symmetric algebra of an ideal or module
• symmetricKernel -- Compute the Rees ring of the image of a matrix
• symmetricPower -- symmetric power
• synonym -- synonym for members of a class
• SYNOPSIS -- a standardized synopsis for use in documentation nodes
• syz -- the syzygy matrix
• syzygyScheme -- construct a syzygy scheme
• table -- make a table from a binary function
• take -- Take some elements from a list or sequence.
• tally -- tally the elements of a list or sequence
• tan -- compute the tangent
• "tangentCone" -- see tangentCone(Ideal)
• tangentSheaf -- tangent sheaf of a projective variety
• tanh -- compute the hyperbolic tangent
• target -- target of a map
• "taskResult" -- see taskResult(Task) -- retrieve the value returned by a task
• temporaryFileName -- make a temporary file name
• tensor -- tensor product
• tensorAssociativity -- associativity isomorphisms for tensor products
• terms -- provide a list of terms of a polynomial
• TEST -- add a test for a package
• testHunekeQuestion -- tests a conjecture on integral closures strengthening the Eisenbud-Mazur conjecture
• tex -- convert to $\TeX$ format
• texMath -- convert to TeX math format
• times -- multiplication
• toAbsolutePath -- the absolute path version of a file name
• toCC -- convert to high-precision complex number
• toDividedPowers -- Translates to divided power monomial basis from ordinary monomial basis
• toDual -- finds the inverse system to an ideal up to a given degree
• toExternalString -- convert to a readable string
• "toField" -- see toField(Ring) -- declare that a ring is a field
• toList -- create a list
• toLower -- convert to lower case
• topCoefficients -- first variable and its coefficient of a polynomial or matrix
• topComponents -- compute top dimensional component of an ideal or module
• toRR -- convert to high-precision real number
• toSequence -- convert to sequence
• toString -- convert to a string
• toUpper -- convert to upper case
• "trace" -- see trace(Matrix) -- trace of a matrix
• transpose -- transpose a table or a matrix
• trim -- minimize generators and relations
• truncate
• truncateOutput
• tutorial -- convert documentation from tutorial format
• ultimate -- ultimate value for an iteration
• "unbag" -- see unbag(Bag)
• uncurry -- uncurry a function
• undocumented -- 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
• unique -- eliminate duplicates from a list
• unsequence -- extract the single element from a sequence of length 1
• unstack -- list the rows of a net
• use -- install or activate object
• userSymbols -- a list of the user's symbols
• utf8 -- encode and decode unicode utf-8-encoded strings
• utf8check (missing documentation)
• value -- evaluate
• values -- values in a hash table
• variety -- get the variety
• vars -- variables
• vector -- make a vector
• versalEmbedding -- Compute a versal embedding
• wait -- wait for child process
• "wedgeProduct" -- see wedgeProduct(ZZ,ZZ,Module) -- the exterior multiplication map
• weightRange -- the pair of lowest and highest weights of the monomials
• whichGm -- Largest Gm satisfied by an ideal
• width -- width of a file or net
• wikipedia -- link to a Wikipedia article
• wrap
• xor -- logical exclusive-or
• youngest -- the youngest member of a sequence
• zero -- whether something is zero
• zeta -- Riemann zeta function

For the programmer

The object Function is a type, with ancestor class Thing.