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HolonomicSystems :: HolonomicSystems

HolonomicSystems -- Examples of Holonomic D-modules

Description

Some examples of D-modules

  • gkz -- create the A-hypergeometric system of Gelfand, Kapranov and Zelevinsky (GKZ)
  • eulerOperators -- Euler Operators
  • toricIdealPartials -- create the toric ideal of an integer matrix
  • AppellF1 -- Appell F1 system of PDE's

Canonical Series Tutorial -- Computing series solutions to regular holonomic systems

  • distraction -- the image in the thetaRing of a torus-fixed element or ideal in a Weyl algebra
  • cssExpts -- the exponents of the canonical series solutions of I in the direction of a weight vector
  • cssExptsMult -- the exponents and multiplicities of the canonical series solutions
  • isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed
  • solveFrobeniusIdeal -- solving Frobenius ideals

Differential Operators

  • diffOps -- differential operators of up to the given order for a quotient polynomial ring

Authors

Version

This documentation describes version 1.0 of HolonomicSystems.

Source code

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

Exports

  • Functions and commands
    • AppellF1 -- Appell F1 system of PDE's
    • createThetaRing (missing documentation)
    • cssExpts -- the exponents of the canonical series solutions of I in the direction of a weight vector
    • cssExptsMult -- the exponents and multiplicities of the canonical series solutions
    • cssLeadTerm -- lead term of the canonical series solutions of I
    • diffOps -- differential operators of up to the given order for a quotient polynomial ring
    • distraction -- the image in the thetaRing of a torus-fixed element or ideal in a Weyl algebra
    • eulerOperators -- Euler Operators
    • gkz -- create the A-hypergeometric system of Gelfand, Kapranov and Zelevinsky (GKZ)
    • indicialIdeal -- the image in the thetaRing of an indicial ideal in a Weyl algebra
    • isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed
    • nilssonSupport (missing documentation)
    • putWeylAlgebra -- transforms output of diffOps into elements of Weyl algebra
    • solveFrobeniusIdeal -- solving Frobenius ideals
    • toricIdealPartials -- create the toric ideal of an integer matrix
    • truncatedCanonicalSeries (missing documentation)
  • Methods
    • "AppellF1(List)" -- see AppellF1 -- Appell F1 system of PDE's
    • "cssExpts(Ideal,List)" -- see cssExpts -- the exponents of the canonical series solutions of I in the direction of a weight vector
    • "cssExptsMult(Ideal,List)" -- see cssExptsMult -- the exponents and multiplicities of the canonical series solutions
    • "cssLeadTerm(Ideal,List)" -- see cssLeadTerm -- lead term of the canonical series solutions of I
    • "diffOps(Ideal,ZZ)" -- see diffOps -- differential operators of up to the given order for a quotient polynomial ring
    • "diffOps(RingElement,ZZ)" -- see diffOps -- differential operators of up to the given order for a quotient polynomial ring
    • "distraction(Ideal,Ring)" -- see distraction -- the image in the thetaRing of a torus-fixed element or ideal in a Weyl algebra
    • "distraction(RingElement,Ring)" -- see distraction -- the image in the thetaRing of a torus-fixed element or ideal in a Weyl algebra
    • "eulerOperators(Matrix,List,PolynomialRing)" -- see eulerOperators -- Euler Operators
    • "eulerOperators(Matrix,PolynomialRing)" -- see eulerOperators -- Euler Operators
    • "gkz(Matrix,List)" -- see gkz -- create the A-hypergeometric system of Gelfand, Kapranov and Zelevinsky (GKZ)
    • "gkz(Matrix,List,PolynomialRing)" -- see gkz -- create the A-hypergeometric system of Gelfand, Kapranov and Zelevinsky (GKZ)
    • "indicialIdeal(Ideal,List)" -- see indicialIdeal -- the image in the thetaRing of an indicial ideal in a Weyl algebra
    • "isTorusFixed(Ideal)" -- see isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed
    • "putWeylAlgebra(HashTable)" -- see putWeylAlgebra -- transforms output of diffOps into elements of Weyl algebra
    • "solveFrobeniusIdeal(Ideal)" -- see solveFrobeniusIdeal -- solving Frobenius ideals
    • "solveFrobeniusIdeal(Ideal,Ring)" -- see solveFrobeniusIdeal -- solving Frobenius ideals
    • "toricIdealPartials(Matrix,PolynomialRing)" -- see toricIdealPartials -- create the toric ideal of an integer matrix
  • Symbols
    • "Vars" -- see AppellF1 -- Appell F1 system of PDE's
    • "BasisElts" -- see diffOps -- differential operators of up to the given order for a quotient polynomial ring
    • "PolyGens" -- see diffOps -- differential operators of up to the given order for a quotient polynomial ring
    • ThetaRing (missing documentation)
    • WtoT (missing documentation)

For the programmer

The object HolonomicSystems is a package.

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