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

Description

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 parametric polynomial
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.

Citation

If you have used this package in your research, please cite it as follows:

@misc{BernsteinSatoSource,
  title = {{BernsteinSato: Bernstein-Sato. Version~1.0}},
  author = {Anton Leykin and Harrison Tsai},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- AnnFs -- differential annihilator of a polynomial in a Weyl algebra
- AnnIFs -- see AnnIFs(Ideal,RingElement) -- the annihilating ideal of f^s for an arbitrary D-module
- bFunction -- b-function
- bFunctionRoots -- see bFunctionRoots(RingElement) -- get roots of a b-function
- Ddual -- holonomic dual of a D-module
- deRham -- deRham cohomology groups for the complement of a hypersurface
- deRhamAll -- see deRhamAll(RingElement) -- deRham complex for the complement of a hypersurface
- DExt -- Ext groups between holonomic modules
- DHom -- D-homomorphisms between holonomic D-modules
- diffRatFun -- derivative of a rational function in a Weyl algebra
- Dintegration -- integration modules of a D-module
- DintegrationAll -- integration modules of a D-module (extended version)
- DintegrationClasses -- integration classes of a D-module
- DintegrationComplex -- derived integration complex of a D-module
- DintegrationIdeal -- integration ideal of a D-module
- Dlocalize -- localization of a D-module
- DlocalizeAll -- localization of a D-module (extended version)
- DlocalizeMap -- localization map from a D-module to its localization
- Dresolution -- resolution of a D-module
- Drestriction -- restriction modules of a D-module
- DrestrictionAll -- restriction modules of a D-module (extended version)
- DrestrictionClasses -- restriction classes of a D-module
- DrestrictionComplex -- derived restriction complex of a D-module
- DrestrictionIdeal -- restriction ideal of a D-module
- ExternalProduct -- external product of modules or complexes
- factorBFunction -- see factorBFunction(RingElement) -- factorization of a b-function
- generalB -- see generalB(List,RingElement) -- global generalized Bernstein-Sato polynomial
- getIntRoots -- see getIntRoots(RingElement) -- get integer roots of a b-function
- globalB -- see globalB(Ideal,RingElement) -- compute global b-function and b-operator for a D-module and a polynomial
- globalBFunction -- see globalBFunction(RingElement) -- global b-function (else known as the Bernstein-Sato polynomial)
- globalBoperator -- see globalBoperator(RingElement) -- compute a b-operator of a polynomial
- hasRationalSing -- see hasRationalSing(List) -- check if a complete intersection has at most rational singularities
- IHmodule -- intersection (co)homology module of an irreducible closed subvariety
- intersectionCohomology -- intersection cohomology of an irreducible affine variety
- isInMultiplierIdeal -- see isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test
- jumpingCoefficients -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
- kappaAnnF1PlanarCurve -- D-annihilator of 1/f for a planar curve
- kOrderAnnFa -- k-th order D-annihilator of a power of a polynomial
- kOrderAnnFs -- see kOrderAnnFa -- k-th order D-annihilator of a power of a polynomial
- lct -- see lct(Ideal) -- compute the log canonical threshold for an ideal
- localBFunction -- see localBFunction(RingElement,Ideal) -- local b-function (a.k.a. the local Bernstein-Sato polynomial)
- localCohom -- local cohomology
- localizeCharacteristicCycle -- the characteristic cycle of the localized $D$-module
- logCohomology -- see logCohomology(RingElement) -- logarithmic cohomology groups in two variables
- multiplierIdeal -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
- paramBpoly -- see paramBpoly(RingElement,File) -- compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial
- polynomialAnnihilator -- annihilator of a polynomial in the Weyl algebra
- polynomialExt -- Ext groups between a holonomic module and a polynomial ring
- polynomialSolutions -- polynomial solutions of a holonomic system
- populateCechComplexCC -- see populateCechComplexCC(Ideal,List) -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules
- pruneCechComplexCC -- see pruneCechComplexCC(MutableHashTable) -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules
- pruneLocalCohom -- see pruneLocalCohom(HashTable) -- prunes local cohomology modules
- rationalFunctionAnnihilator -- annihilator of a rational function in Weyl algebra
- rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionSolutions -- rational solutions of a holonomic system
- reiffen -- Reiffen's curve
- WeylClosure -- Weyl closure of an ideal
Methods
- AnnFs(RingElement) -- see AnnFs -- differential annihilator of a polynomial in a Weyl algebra
- AnnIFs(Ideal,RingElement) -- the annihilating ideal of f^s for an arbitrary D-module
- bFunction(Ideal,List) -- b-function of an ideal
- bFunction(Module,List,List) -- b-function of a holonomic D-module
- bFunctionRoots(RingElement) -- get roots of a b-function
- Ddual(Ideal) -- see Ddual -- holonomic dual of a D-module
- Ddual(Module) -- see Ddual -- holonomic dual of a D-module
- deRham(RingElement) -- see deRham -- deRham cohomology groups for the complement of a hypersurface
- deRham(ZZ,RingElement) -- see deRham -- deRham cohomology groups for the complement of a hypersurface
- deRham(Ideal) (missing documentation)
- deRham(ZZ,Ideal) (missing documentation)
- deRhamAll(RingElement) -- deRham complex for the complement of a hypersurface
- DExt(Module,Module) -- see DExt -- Ext groups between holonomic modules
- DExt(Module,Module,List) -- see DExt -- Ext groups between holonomic modules
- DHom(Ideal,Ideal) -- see DHom -- D-homomorphisms between holonomic D-modules
- DHom(Module,Module) -- see DHom -- D-homomorphisms between holonomic D-modules
- DHom(Module,Module,List) -- see DHom -- D-homomorphisms between holonomic D-modules
- diffRatFun(List,RingElement) -- see diffRatFun -- derivative of a rational function in a Weyl algebra
- diffRatFun(List,RingElement,RingElement,ZZ) -- see diffRatFun -- derivative of a rational function in a Weyl algebra
- Dintegration(Ideal,List) -- see Dintegration -- integration modules of a D-module
- Dintegration(Module,List) -- see Dintegration -- integration modules of a D-module
- Dintegration(ZZ,Ideal,List) -- see Dintegration -- integration modules of a D-module
- Dintegration(ZZ,Module,List) -- see Dintegration -- integration modules of a D-module
- DintegrationAll(Ideal,List) -- see DintegrationAll -- integration modules of a D-module (extended version)
- DintegrationAll(Module,List) -- see DintegrationAll -- integration modules of a D-module (extended version)
- DintegrationClasses(Ideal,List) -- see DintegrationClasses -- integration classes of a D-module
- DintegrationClasses(Module,List) -- see DintegrationClasses -- integration classes of a D-module
- DintegrationClasses(ZZ,Ideal,List) -- see DintegrationClasses -- integration classes of a D-module
- DintegrationClasses(ZZ,Module,List) -- see DintegrationClasses -- integration classes of a D-module
- DintegrationComplex(Ideal,List) -- see DintegrationComplex -- derived integration complex of a D-module
- DintegrationComplex(Module,List) -- see DintegrationComplex -- derived integration complex of a D-module
- DintegrationIdeal(Ideal,List) -- see DintegrationIdeal -- integration ideal of a D-module
- Dlocalize(Ideal,RingElement) -- see Dlocalize -- localization of a D-module
- Dlocalize(Module,RingElement) -- see Dlocalize -- localization of a D-module
- DlocalizeAll(Ideal,RingElement) -- see DlocalizeAll -- localization of a D-module (extended version)
- DlocalizeAll(Module,RingElement) -- see DlocalizeAll -- localization of a D-module (extended version)
- DlocalizeMap(Ideal,RingElement) -- see DlocalizeMap -- localization map from a D-module to its localization
- DlocalizeMap(Module,RingElement) -- see DlocalizeMap -- localization map from a D-module to its localization
- Dresolution(Ideal) -- see Dresolution -- resolution of a D-module
- Dresolution(Ideal,List) -- see Dresolution -- resolution of a D-module
- Dresolution(Module) -- see Dresolution -- resolution of a D-module
- Dresolution(Module,List) -- see Dresolution -- resolution of a D-module
- Drestriction(Ideal,List) -- see Drestriction -- restriction modules of a D-module
- Drestriction(Module,List) -- see Drestriction -- restriction modules of a D-module
- Drestriction(ZZ,Ideal,List) -- see Drestriction -- restriction modules of a D-module
- Drestriction(ZZ,Module,List) -- see Drestriction -- restriction modules of a D-module
- DrestrictionAll(Ideal,List) -- see DrestrictionAll -- restriction modules of a D-module (extended version)
- DrestrictionAll(Module,List) -- see DrestrictionAll -- restriction modules of a D-module (extended version)
- DrestrictionClasses(Ideal,List) -- see DrestrictionClasses -- restriction classes of a D-module
- DrestrictionClasses(Module,List) -- see DrestrictionClasses -- restriction classes of a D-module
- DrestrictionClasses(ZZ,Ideal,List) -- see DrestrictionClasses -- restriction classes of a D-module
- DrestrictionClasses(ZZ,Module,List) -- see DrestrictionClasses -- restriction classes of a D-module
- DrestrictionComplex(Ideal,List) -- see DrestrictionComplex -- derived restriction complex of a D-module
- DrestrictionComplex(Module,List) -- see DrestrictionComplex -- derived restriction complex of a D-module
- DrestrictionIdeal(Ideal,List) -- see DrestrictionIdeal -- restriction ideal of a D-module
- ExternalProduct(ChainComplex,ChainComplex) -- see ExternalProduct -- external product of modules or complexes
- ExternalProduct(Module,Module) -- see ExternalProduct -- external product of modules or complexes
- factorBFunction(RingElement) -- factorization of a b-function
- generalB(List) -- see generalB(List,RingElement) -- global generalized Bernstein-Sato polynomial
- generalB(List,RingElement) -- global generalized Bernstein-Sato polynomial
- getIntRoots(RingElement) -- get integer roots of a b-function
- globalB(Ideal,RingElement) -- compute global b-function and b-operator for a D-module and a polynomial
- globalBFunction(RingElement) -- global b-function (else known as the Bernstein-Sato polynomial)
- globalBoperator(RingElement) -- compute a b-operator of a polynomial
- hasRationalSing(List) -- check if a complete intersection has at most rational singularities
- IHmodule(Ideal) -- see IHmodule -- intersection (co)homology module of an irreducible closed subvariety
- intersectionCohomology(Ideal) -- see intersectionCohomology -- intersection cohomology of an irreducible affine variety
- intersectionCohomology(ZZ,Ideal) -- see intersectionCohomology -- intersection cohomology of an irreducible affine variety
- isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test
- jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
- jumpingCoefficients(Ideal,QQ,QQ) -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
- jumpingCoefficients(Ideal,QQ,ZZ) -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
- jumpingCoefficients(Ideal,ZZ,QQ) -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
- jumpingCoefficients(Ideal,ZZ,ZZ) -- see jumpingCoefficients(Ideal) -- jumping coefficients and corresponding multiplier ideals
- kappaAnnF1PlanarCurve(RingElement) -- see kappaAnnF1PlanarCurve -- D-annihilator of 1/f for a planar curve
- kOrderAnnFa(ZZ,RingElement,ZZ) -- see kOrderAnnFa -- k-th order D-annihilator of a power of a polynomial
- kOrderAnnFs(ZZ,RingElement) -- see kOrderAnnFa -- k-th order D-annihilator of a power of a polynomial
- lct(Ideal) -- compute the log canonical threshold for an ideal
- localBFunction(RingElement,Ideal) -- local b-function (a.k.a. the local Bernstein-Sato polynomial)
- localCohom(Ideal) -- local cohomology of a polynomial ring
- localCohom(Ideal,Module) -- local cohomology of a D-module
- localCohom(List,Ideal) -- local cohomology of a polynomial ring
- localCohom(List,Ideal,Module) -- local cohomology of a D-module
- localCohom(ZZ,Ideal) -- local cohomology of a polynomial ring
- localCohom(ZZ,Ideal,Module) -- local cohomology of a D-module
- localizeCharacteristicCycle(Ideal,RingElement) -- see localizeCharacteristicCycle -- the characteristic cycle of the localized $D$-module
- localizeCharacteristicCycle(List,RingElement) -- see localizeCharacteristicCycle -- the characteristic cycle of the localized $D$-module
- logCohomology(RingElement) -- logarithmic cohomology groups in two variables
- multiplierIdeal(Ideal,List) -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
- multiplierIdeal(Ideal,QQ) -- multiplier ideal
- multiplierIdeal(Ideal,ZZ) -- see multiplierIdeal(Ideal,QQ) -- multiplier ideal
- paramBpoly(RingElement,File) -- compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial
- polynomialAnnihilator(RingElement) -- see polynomialAnnihilator -- annihilator of a polynomial in the Weyl algebra
- polynomialExt(Ideal) -- see polynomialExt -- Ext groups between a holonomic module and a polynomial ring
- polynomialExt(Module) -- see polynomialExt -- Ext groups between a holonomic module and a polynomial ring
- polynomialExt(ZZ,Ideal) -- see polynomialExt -- Ext groups between a holonomic module and a polynomial ring
- polynomialExt(ZZ,Module) -- see polynomialExt -- Ext groups between a holonomic module and a polynomial ring
- polynomialSolutions(Ideal) -- see polynomialSolutions -- polynomial solutions of a holonomic system
- polynomialSolutions(Ideal,List) -- see polynomialSolutions -- polynomial solutions of a holonomic system
- polynomialSolutions(Module) -- see polynomialSolutions -- polynomial solutions of a holonomic system
- polynomialSolutions(Module,List) -- see polynomialSolutions -- polynomial solutions of a holonomic system
- populateCechComplexCC(Ideal,List) -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules
- pruneCechComplexCC(MutableHashTable) -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules
- pruneLocalCohom(HashTable) -- prunes local cohomology modules
- rationalFunctionAnnihilator(RingElement) -- see rationalFunctionAnnihilator -- annihilator of a rational function in Weyl algebra
- rationalFunctionAnnihilator(RingElement,RingElement) -- see rationalFunctionAnnihilator -- annihilator of a rational function in Weyl algebra
- rationalFunctionExt(Ideal) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionExt(Ideal,RingElement) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionExt(Module) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionExt(Module,RingElement) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionExt(ZZ,Ideal) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionExt(ZZ,Ideal,RingElement) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionExt(ZZ,Module) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionExt(ZZ,Module,RingElement) -- see rationalFunctionExt -- Ext(holonomic D-module, polynomial ring localized at the singular locus)
- rationalFunctionSolutions(Ideal) -- see rationalFunctionSolutions -- rational solutions of a holonomic system
- rationalFunctionSolutions(Ideal,List) -- see rationalFunctionSolutions -- rational solutions of a holonomic system
- rationalFunctionSolutions(Ideal,List,List) -- see rationalFunctionSolutions -- rational solutions of a holonomic system
- rationalFunctionSolutions(Ideal,RingElement) -- see rationalFunctionSolutions -- rational solutions of a holonomic system
- rationalFunctionSolutions(Ideal,RingElement,List) -- see rationalFunctionSolutions -- rational solutions of a holonomic system
- reiffen(ZZ,ZZ) -- see reiffen -- Reiffen's curve
- WeylClosure(Ideal) -- see WeylClosure -- Weyl closure of an ideal
- WeylClosure(Ideal,RingElement) -- see WeylClosure -- Weyl closure of an ideal
Symbols
- Alg
- BFunction -- a key in the hashtable created by Drestriction/Dintegration
- IntRing -- see bFunction(...,Strategy=>...) -- specify strategy for computing b-function
- NonGeneric -- see bFunction(...,Strategy=>...) -- specify strategy for computing b-function
- TryGeneric -- see bFunction(...,Strategy=>...) -- specify strategy for computing b-function
- Boperator -- a key attached by globalB and Dlocalize
- Boundaries -- a key in the hashtable created by Drestriction/Dintegration
- Bpolynomial -- a key attached by globalB
- CohomologyGroups -- a key in the hashtable created by deRham
- Cycles -- a key in the hashtable created by Drestriction/Dintegration
- Oaku -- see Dlocalize(...,Strategy=>...) -- strategy for computing a localization of a D-module
- OTW -- see Dlocalize(...,Strategy=>...) -- strategy for computing a localization of a D-module
- OTWcyclic -- see Dlocalize(...,Strategy=>...) -- strategy for computing a localization of a D-module
- annFS -- see DlocalizeAll -- localization of a D-module (extended version)
- Bfunction -- see DlocalizeAll -- localization of a D-module (extended version)
- GeneratorPower -- see DlocalizeAll -- localization of a D-module (extended version)
- IntegrateBfunction -- see DlocalizeAll -- localization of a D-module (extended version)
- LocMap -- see DlocalizeAll -- localization of a D-module (extended version)
- LocModule -- see DlocalizeAll -- localization of a D-module (extended version)
- Duality -- an option for polynomialSolutions=>Alg
- Explicit -- a key in the hashtable created by Drestriction/Dintegration
- Exponents -- a key in the hashtable created by Drestriction/Dintegration
- GD -- an option for polynomialSolutions=>Alg
- GenCycles -- a key in the hashtable created by Drestriction/Dintegration
- Exponent -- see generalB(...,Exponent=>...) -- specify exponent m for m-generalized Bernstein-Sato polynomial
- InitialIdeal -- see generalB(...,Strategy=>...) -- specify strategy for computing generalized Bernstein-Sato polynomial
- StarIdeal -- see generalB(...,Strategy=>...) -- specify strategy for computing generalized Bernstein-Sato polynomial
- GeneralBernsteinSato -- a strategy option for lct, globalBFunction
- ReducedB -- see globalBFunction(...,Strategy=>...) -- specify strategy for computing global b-function
- ViaAnnFs -- see globalBFunction(...,Strategy=>...) -- specify strategy for computing global b-function
- HomologyModules -- a key in a hashtable; an option of DExt
- LocCohomStrategy -- see IHmodule -- intersection (co)homology module of an irreducible closed subvariety
- Info
- IntegrateComplex -- a key in the hashtable created by Dintegration
- generalizedBFunction -- see isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test
- mGeneralizedBFunction -- see isInMultiplierIdeal(RingElement,Ideal,QQ) -- multiplier ideal membership test
- LocalizeMap -- a key in the hashtable created by deRham
- LocStrategy
- None -- an option for DExt=>Special
- OaTa -- an option for localCohom=>Strategy
- OaTaWa -- an option for localCohom => LocStrategy
- OmegaRes -- a key in the hashtable created by deRham
- Output
- PreCycles -- a key in the hashtable created by deRham
- projMap1 -- a key attached by ExternalProduct
- projMap2 -- a key attached by ExternalProduct
- Schreyer -- strategy for computing a resolution of a D-module
- Special
- TransferCycles -- a key in the hashtable created by deRham
- twistInvMap -- a key attached by ExternalProduct
- TwistMap -- indicates whether TwistMap should be computed
- twistMap -- a key attached by ExternalProduct
- Vhomogenize -- strategy for computing a resolution of a D-module
- ViaBFunction -- a strategy option for lct
- ViaColonIdeal (missing documentation)
- ViaElimination (missing documentation)
- ViaLinearAlgebra -- an option for generalB=>Strategy
- VResolution -- a key in the hashtable created by Drestriction/Dintegration
- Walther -- an option for localCohom=>Strategy
Other things
- IH -- intersection cohomology of an irreducible affine variety

For the programmer

The object BernsteinSato is a package, defined in BernsteinSato.m2, with auxiliary files in BernsteinSato/.

The source of this document is in BernsteinSato/DOC/main.m2:52:0.