# Dmodules -- algorithms for D-modules

## Description

### 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

### 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 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

## Authors

• Anton Leykin
• Harrison Tsai

## Version

This documentation describes version 1.4.0.1 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/.