TriangularSets -- triangular decompositions of polynomial ideals

Description

This package allows to decompose polynomial ideals into triangular sets

i1 : R = QQ[a..h, MonomialOrder=>Lex];

i2 : I = ideal {a*d - b*c, c*f - d*e, e*h - f*g};

o2 : Ideal of R

i3 : triangularize I

o3 = {{c, d, f, h}, {b, d, e, f}, {a*d - b*c, e, f} / d, {a*d - b*c, c*f -
     ------------------------------------------------------------------------
     d*e, e*h - f*g} / {d, f, h}, {c, d, e*h - f*g} / h, {c, d, e, f}, {a*d -
     ------------------------------------------------------------------------
     b*c, c*f - d*e, g, h} / {d, f}, {b, d, f, h}, {c, d, g, h}}

o3 : List

The method triangularize is implemented in M2 only for monomial and binomial ideals. For the general case we interface to Maple.

This package also provides methods for manipulating triangular sets:

dim(TriaSystem) -- dimension of a triangular set
saturate(TriaSystem) -- saturated ideal of a triangular system
isRegularChain(TriaSystem) -- whether a triangular set is a regular chain
isStronglyNormalized(TriaSystem) -- whether a triangular set is strongly normalized
RingElement % TriaSystem -- pseudo-remainder by a triangular set
resultant(RingElement,TriaSystem)

References

P. Aubry, D. Lazard, M. Moreno Maza (1999), "On the theories of triangular sets", in "Journal of Symbolic Computation", 28(1):105-–124
E. Hubert (2003), "Notes on triangular sets and triangulation-decomposition algorithms I: Polynomial systems", in "Springer"
M. Kalkbrener (1993), "A generalized Euclidean algorithm for computing triangular representations of algebraic varieties", in "Journal of Symbolic Computation", 15(2):143-–167
D. Lazard (1992), "Solving zero-dimensional algebraic systems", in "Journal of Symbolic Computation", 13(2):117-–131
D. Wang (2001), "Elimination methods", in "Springer Science & Business Media"

Author

Diego Cifuentes <[email protected]>

Version

This documentation describes version 0.1 of TriangularSets.

Source code

The source code from which this documentation is derived is in the file TriangularSets.m2. The auxiliary files accompanying it are in the directory TriangularSets/.

Exports

Types
- TriaSystem -- a triangular system
Functions and commands
- checkInterface -- whether the Maple interface is working (for developers)
- checkTriangularize -- tests method "triangularize" (for developers)
- freeVariables -- free variables of a triangular set
- gbList -- list of generators of a Groebner basis
- ineqs -- inequations of a triangular system
- initial -- initial of a polynomial
- isPrimeSimple -- simple primality test of triangular systems
- isRegularChain -- whether a triangular set is a regular chain
- isStronglyNormalized -- whether a triangular set is strongly normalized
- minimalObjects -- minimal elements in a partial order
- mvar -- main variable of a polynomial
- triangularize -- triangular decomposition of polynomial systems
- triangularizeBatch -- triangularize a batch of polynomial systems
- triaSystem -- a triangular system
Methods
- degree(TriaSystem) -- product of the degrees in a triangular set
- codim(TriaSystem) -- see dim(TriaSystem) -- dimension of a triangular set
- dim(TriaSystem) -- dimension of a triangular set
- freeVariables(TriaSystem) -- see freeVariables -- free variables of a triangular set
- generators(TriaSystem) -- equations of a triangular system
- ineqs(TriaSystem) -- see ineqs -- inequations of a triangular system
- initial(RingElement) -- see initial -- initial of a polynomial
- initial(TriaSystem) -- see initial -- initial of a polynomial
- isPrimeSimple(TriaSystem) -- see isPrimeSimple -- simple primality test of triangular systems
- isRegularChain(TriaSystem) -- see isRegularChain -- whether a triangular set is a regular chain
- isStronglyNormalized(TriaSystem) -- see isStronglyNormalized -- whether a triangular set is strongly normalized
- mvar(RingElement) -- see mvar -- main variable of a polynomial
- pseudoRemainder(RingElement,RingElement,RingElement) -- pseudo-remainder
- resultant(RingElement,TriaSystem) -- iterated resultant by a triangular set
- pseudoRemainder(RingElement,TriaSystem) -- see RingElement % TriaSystem -- pseudo-remainder by a triangular set
- RingElement % TriaSystem -- pseudo-remainder by a triangular set
- RingMap TriaSystem -- apply ring map to a triangular system
- saturate(TriaSystem) -- saturated ideal of a triangular system
- triangularize(Ideal) -- see triangularize -- triangular decomposition of polynomial systems
- triangularize(MonomialIdeal) -- see triangularize -- triangular decomposition of polynomial systems
- triangularize(Ring,List) -- see triangularize -- triangular decomposition of polynomial systems
- triangularize(Ring,List,List) -- see triangularize -- triangular decomposition of polynomial systems
- net(TriaSystem) -- see TriaSystem -- a triangular system
- triaSystem(Ring,List) -- see triaSystem -- a triangular system
- triaSystem(Ring,List,List) -- see triaSystem -- a triangular system
Symbols
- TriangularDecompAlgorithm -- possible values: "Monomial", "Binomial", "Maple", "Epsilon"

For the programmer

The object TriangularSets is a package.