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Dmodules -- algorithms for D-modules

Description

How to make Weyl algebras:

Basic commands:

  • gbw -- Groebner basis w.r.t. a weight
  • inw -- initial form/ideal w.r.t. a weight
  • Fourier -- Fourier transform for Weyl algebra
  • Dtransposition -- standard transposition for Weyl algebra
  • makeCyclic -- finds a cyclic generator of a D-module
  • makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring
  • stafford -- computes 2 generators for a given ideal in the Weyl algebra

Some examples of D-modules:

  • gkz -- GKZ A-hypergeometric ideal
  • AppellF1 -- Appell F1 system of PDE's
  • PolyAnn -- annihilator of a polynomial in Weyl algebra
  • RatAnn -- annihilator of a rational function in Weyl algebra

Basic invariants of D-modules:

B-functions:

  • bFunction -- b-function
  • globalBFunction -- global b-function (else known as the Bernstein-Sato polynomial)
  • globalB -- compute global b-function and b-operator for a D-module and a polynomial
  • globalBoperator -- compute a b-operator of a polynomial
  • generalB -- global general Bernstein-Sato polynomial
  • paramBpoly -- compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients
  • factorBFunction -- factorization of a b-function
  • bFunctionRoots -- get roots of a b-function
  • getIntRoots -- get integer roots of a b-function
  • AnnFs -- annihilator ideal of fs
  • AnnIFs -- annihilator ideal of fs for an arbitrary D-module

Resolutions and Functors:

  • Dresolution -- resolution of a D-module
  • Dlocalize -- localization of a D-module
  • WeylClosure -- Weyl closure of an ideal
  • Ddual -- holonomic dual of a D-module
  • Drestriction -- restriction modules of a D-module
  • Dintegration -- integration modules of a D-module
  • DHom -- D-homomorphisms between holonomic D-modules
  • DExt -- Ext groups between holonomic modules
  • PolyExt -- Ext groups between a holonomic module and a polynomial ring
  • RatExt -- Ext(holonomic D-module, polynomial ring localized at the sigular locus)

Applications:

  • localCohom -- local cohomology
  • deRham -- deRham cohomology groups for the complement of a hypersurface
  • PolySols -- polynomial solutions of a holonomic system
  • RatSols -- rational solutions of a holonomic system
  • diffOps -- differential operators of up to the given order for a quotient polynomial ring
  • populateCechComplexCC, pruneCechComplexCC -- characteristic cycles of local cohomology
  • logCohomology -- logarithmic cohomology groups in two variables
  • lct -- compute the log canonical threshold for an ideal

Programming aids:

  • createDpairs -- pairs up the variables in Weyl algebra
  • Dtrace -- set the depth of comments made by D-module routines
  • setHomSwitch -- toggles the use of homogeneous Weyl algebra

Authors

Version

This documentation describes version 1.1 of Dmodules.

Source code

The source code is in the file Dmodules.m2.

Exports