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TriangularSets :: TriangularSets

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

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:

References

Author

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
  • Functions and commands
  • 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
    • 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

For the programmer

The object TriangularSets is a package.