BernsteinSato -- algorithms for b-functions, local cohomology, and intersection cohomology

B-functions

• bFunction -- b-function
• generalB -- global generalized Bernstein-Sato polynomial
• globalB -- compute global b-function and b-operator for a D-module and a polynomial
• globalBFunction -- global b-function (else known as the Bernstein-Sato polynomial)
• globalBoperator -- compute a b-operator of a polynomial
• localBFunction -- local b-function (a.k.a. the local Bernstein-Sato polynomial)
• factorBFunction -- factorization of a b-function
• bFunctionRoots -- get roots of a b-function
• getIntRoots -- get integer roots of a b-function
• paramBpoly -- compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients
• AnnFs -- differential annihilator of a polynomial in a Weyl algebra
• AnnIFs -- the annihilating 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
• polynomialExt -- Ext groups between a holonomic module and a polynomial ring
• rationalFunctionExt -- 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
• polynomialSolutions -- polynomial solutions of a holonomic system
• rationalFunctionSolutions -- rational solutions of a holonomic system
• populateCechComplexCC -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules
• pruneCechComplexCC -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules
• logCohomology -- logarithmic cohomology groups in two variables
• Multiplier Ideals
  • lct -- compute the log canonical threshold for an ideal
  • multiplierIdeal -- multiplier ideal
  • isInMultiplierIdeal -- multiplier ideal membership test
  • jumpingCoefficients -- jumping coefficients and corresponding multiplier ideals
  • hasRationalSing -- check if a complete intersection has at most rational singularities

Authors

• Anton Leykin <[email protected]>
• Harrison Tsai

Version

This documentation describes version 1.0 of BernsteinSato.

Source code

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

Exports