Dmodules -- algorithms for D-modules

Description

To begin, read the D-modules tutorial.

How to make Weyl algebras:

• WeylAlgebra -- an optional argument
• makeWeylAlgebra -- Weyl algebra corresponding to a polynomial ring

Basic commands:

• gbw -- Groebner bases 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
• stafford -- computes 2 generators for a given ideal in the Weyl algebra

Some examples of D-modules:

• gkz -- The A-hypergeometric systems of Gelfand, Kapranov and Zelevinsky (GKZ)
• AppellF1 -- Appell F1 system of PDE's
• PolyAnn -- annihilator of a polynomial in the Weyl algebra
• RatAnn -- annihilator of a rational function in Weyl algebra

Basic invariants of D-modules:

• Ddim -- dimension of a D-module
• holonomicRank -- rank of a D-module
• charIdeal -- characteristic ideal of a D-module
• singLocus -- singular locus of a D-module

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 f^s
• AnnIFs -- annihilator ideal of f^s 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 singular 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
• multiplierIdeal, isInMultiplierIdeal, jumpingCoefficients -- multiplier ideals
• hasRationalSing -- check if a complete intersection has at most rational singularities

Canonical Series:

• distraction -- the image in the thetaRing of a torus-fixed ideal in a Weyl algebra
• genToDistractionGens -- the image in the thetaRing of a torus-fixed element 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 of I in the direction of a weight vector
• isTorusFixed -- checks if an ideal in a Weyl algebra is torus-fixed
• Canonical Series Tutorial -- Computing series solutions to regular holonomic systems

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