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Dmodules :: Dmodules

Dmodules -- algorithms for D-modules


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:


  • 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 generalized Bernstein-Sato polynomial
  • localBFunction -- local b-function (a.k.a. the local 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)


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



This documentation describes version of Dmodules.

Source code

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