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<!DOCTYPE html> <html lang="en"> <head> <title>BernsteinSato -- algorithms for b-functions, local cohomology, and intersection cohomology</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", 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href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="algorithms for b-functions, local cohomology, and intersection cohomology" href="index.html">BernsteinSato</a> :: <a title="algorithms for b-functions, local cohomology, and intersection cohomology" href="index.html">BernsteinSato</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="___Alg.html">next</a> | previous | <a href="___Alg.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>BernsteinSato -- algorithms for b-functions, local cohomology, and intersection cohomology</h1> <div> <h2>Description</h2> <div> <h4>B-functions</h4> <ul> <li><span><a title="b-function" href="_b__Function.html">bFunction</a> -- b-function</span></li> <li><span><a title="global generalized Bernstein-Sato polynomial" href="_general__B_lp__List_cm__Ring__Element_rp.html">generalB</a> -- global generalized Bernstein-Sato polynomial</span></li> <li><span><a title="compute global b-function and b-operator for a D-module and a polynomial" href="_global__B_lp__Ideal_cm__Ring__Element_rp.html">globalB</a> -- compute global b-function and b-operator for a D-module and a polynomial</span></li> <li><span><a title="global b-function (else known as the Bernstein-Sato polynomial)" href="_global__B__Function_lp__Ring__Element_rp.html">globalBFunction</a> -- global b-function (else known as the Bernstein-Sato polynomial)</span></li> <li><span><a title="compute a b-operator of a polynomial" href="_global__Boperator_lp__Ring__Element_rp.html">globalBoperator</a> -- compute a b-operator of a polynomial</span></li> <li><span><a title="local b-function (a.k.a. the local Bernstein-Sato polynomial)" href="_local__B__Function_lp__Ring__Element_cm__Ideal_rp.html">localBFunction</a> -- local b-function (a.k.a. the local Bernstein-Sato polynomial)</span></li> <li><span><a title="factorization of a b-function" href="_factor__B__Function_lp__Ring__Element_rp.html">factorBFunction</a> -- factorization of a b-function</span></li> <li><span><a title="get roots of a b-function" href="_b__Function__Roots_lp__Ring__Element_rp.html">bFunctionRoots</a> -- get roots of a b-function</span></li> <li><span><a title="get integer roots of a b-function" href="_get__Int__Roots_lp__Ring__Element_rp.html">getIntRoots</a> -- get integer roots of a b-function</span></li> <li><span><a title="compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial" href="_param__Bpoly_lp__Ring__Element_cm__File_rp.html">paramBpoly</a> -- compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial</span></li> <li><span><a title="differential annihilator of a polynomial in a Weyl algebra" href="___Ann__Fs.html">AnnFs</a> -- differential annihilator of a polynomial in a Weyl algebra</span></li> <li><span><a title="the annihilating ideal of f^s for an arbitrary D-module" href="___Ann__I__Fs_lp__Ideal_cm__Ring__Element_rp.html">AnnIFs</a> -- the annihilating ideal of f^s for an arbitrary D-module</span></li> </ul> <h4>Resolutions and Functors</h4> <ul> <li><span><a title="resolution of a D-module" href="___Dresolution.html">Dresolution</a> -- resolution of a D-module</span></li> <li><span><a title="localization of a D-module" href="___Dlocalize.html">Dlocalize</a> -- localization of a D-module</span></li> <li><span><a title="Weyl closure of an ideal" href="___Weyl__Closure.html">WeylClosure</a> -- Weyl closure of an ideal</span></li> <li><span><a title="holonomic dual of a D-module" href="___Ddual.html">Ddual</a> -- holonomic dual of a D-module</span></li> <li><span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></li> <li><span><a title="integration modules of a D-module" href="___Dintegration.html">Dintegration</a> -- integration modules of a D-module</span></li> <li><span><a title="D-homomorphisms between holonomic D-modules" href="___D__Hom.html">DHom</a> -- D-homomorphisms between holonomic D-modules</span></li> <li><span><a title="Ext groups between holonomic modules" href="___D__Ext.html">DExt</a> -- Ext groups between holonomic modules</span></li> <li><span><a title="Ext groups between a holonomic module and a polynomial ring" href="_polynomial__Ext.html">polynomialExt</a> -- Ext groups between a holonomic module and a polynomial ring</span></li> <li><span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></li> </ul> <h4>Applications:</h4> <ul> <li><span><a title="local cohomology" href="_local__Cohom.html">localCohom</a> -- local cohomology</span></li> <li><span><a title="deRham cohomology groups for the complement of a hypersurface" href="_de__Rham.html">deRham</a> -- deRham cohomology groups for the complement of a hypersurface</span></li> <li><span><a title="polynomial solutions of a holonomic system" href="_polynomial__Solutions.html">polynomialSolutions</a> -- polynomial solutions of a holonomic system</span></li> <li><span><a title="rational solutions of a holonomic system" href="_rational__Function__Solutions.html">rationalFunctionSolutions</a> -- rational solutions of a holonomic system</span></li> <li><span><a title="Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules" href="_populate__Cech__Complex__C__C_lp__Ideal_cm__List_rp.html">populateCechComplexCC</a> -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules</span></li> <li><span><a title="reduction of the Cech complex that produces characteristic cycles of local cohomology modules" href="_prune__Cech__Complex__C__C_lp__Mutable__Hash__Table_rp.html">pruneCechComplexCC</a> -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules</span></li> <li><span><a title="logarithmic cohomology groups in two variables" href="_log__Cohomology_lp__Ring__Element_rp.html">logCohomology</a> -- logarithmic cohomology groups in two variables</span></li> <li>Multiplier Ideals <ul> <li><span><a title="compute the log canonical threshold for an ideal" href="_lct_lp__Ideal_rp.html">lct</a> -- compute the log canonical threshold for an ideal</span></li> <li><span><a title="multiplier ideal" href="_multiplier__Ideal_lp__Ideal_cm__Q__Q_rp.html">multiplierIdeal</a> -- multiplier ideal</span></li> <li><span><a title="multiplier ideal membership test" href="_is__In__Multiplier__Ideal_lp__Ring__Element_cm__Ideal_cm__Q__Q_rp.html">isInMultiplierIdeal</a> -- multiplier ideal membership test</span></li> <li><span><a title="jumping coefficients and corresponding multiplier ideals" href="_jumping__Coefficients_lp__Ideal_rp.html">jumpingCoefficients</a> -- jumping coefficients and corresponding multiplier ideals</span></li> <li><span><a title="check if a complete intersection has at most rational singularities" href="_has__Rational__Sing_lp__List_rp.html">hasRationalSing</a> -- check if a complete intersection has at most rational singularities</span></li> </ul> </li> </ul> </div> </div> <div> <h3>Menu</h3> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li>Anton Leykin<span> <<a href="mailto:leykin%40math.gatech.edu">leykin@math.gatech.edu</a>></span></li> <li>Harrison Tsai</li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.0</b> of BernsteinSato, released <b>February 2023</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@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/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><a title="differential annihilator of a polynomial in a Weyl algebra" href="___Ann__Fs.html">AnnFs</a> -- differential annihilator of a polynomial in a Weyl algebra</span></li> <li><span><kbd>AnnIFs</kbd> -- see <span><a title="the annihilating ideal of f^s for an arbitrary D-module" href="___Ann__I__Fs_lp__Ideal_cm__Ring__Element_rp.html">AnnIFs(Ideal,RingElement)</a> -- the annihilating ideal of f^s for an arbitrary D-module</span></span></li> <li><span><a title="b-function" href="_b__Function.html">bFunction</a> -- b-function</span></li> <li><span><kbd>bFunctionRoots</kbd> -- see <span><a title="get roots of a b-function" href="_b__Function__Roots_lp__Ring__Element_rp.html">bFunctionRoots(RingElement)</a> -- get roots of a b-function</span></span></li> <li><span><a title="holonomic dual of a D-module" href="___Ddual.html">Ddual</a> -- holonomic dual of a D-module</span></li> <li><span><a title="deRham cohomology groups for the complement of a hypersurface" href="_de__Rham.html">deRham</a> -- deRham cohomology groups for the complement of a hypersurface</span></li> <li><span><kbd>deRhamAll</kbd> -- see <span><a title="deRham complex for the complement of a hypersurface" href="_de__Rham__All_lp__Ring__Element_rp.html">deRhamAll(RingElement)</a> -- deRham complex for the complement of a hypersurface</span></span></li> <li><span><a title="Ext groups between holonomic modules" href="___D__Ext.html">DExt</a> -- Ext groups between holonomic modules</span></li> <li><span><a title="D-homomorphisms between holonomic D-modules" href="___D__Hom.html">DHom</a> -- D-homomorphisms between holonomic D-modules</span></li> <li><span><a title="derivative of a rational function in a Weyl algebra" href="_diff__Rat__Fun.html">diffRatFun</a> -- derivative of a rational function in a Weyl algebra</span></li> <li><span><a title="integration modules of a D-module" href="___Dintegration.html">Dintegration</a> -- integration modules of a D-module</span></li> <li><span><a title="integration modules of a D-module (extended version)" href="___Dintegration__All.html">DintegrationAll</a> -- integration modules of a D-module (extended version)</span></li> <li><span><a title="integration classes of a D-module" href="___Dintegration__Classes.html">DintegrationClasses</a> -- integration classes of a D-module</span></li> <li><span><a title="derived integration complex of a D-module" href="___Dintegration__Complex.html">DintegrationComplex</a> -- derived integration complex of a D-module</span></li> <li><span><a title="integration ideal of a D-module" href="___Dintegration__Ideal.html">DintegrationIdeal</a> -- integration ideal of a D-module</span></li> <li><span><a title="localization of a D-module" href="___Dlocalize.html">Dlocalize</a> -- localization of a D-module</span></li> <li><span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></li> <li><span><a title="localization map from a D-module to its localization" href="___Dlocalize__Map.html">DlocalizeMap</a> -- localization map from a D-module to its localization</span></li> <li><span><a title="resolution of a D-module" href="___Dresolution.html">Dresolution</a> -- resolution of a D-module</span></li> <li><span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></li> <li><span><a title="restriction modules of a D-module (extended version)" href="___Drestriction__All.html">DrestrictionAll</a> -- restriction modules of a D-module (extended version)</span></li> <li><span><a title="restriction classes of a D-module" href="___Drestriction__Classes.html">DrestrictionClasses</a> -- restriction classes of a D-module</span></li> <li><span><a title="derived restriction complex of a D-module" href="___Drestriction__Complex.html">DrestrictionComplex</a> -- derived restriction complex of a D-module</span></li> <li><span><a title="restriction ideal of a D-module" href="___Drestriction__Ideal.html">DrestrictionIdeal</a> -- restriction ideal of a D-module</span></li> <li><span><a title="external product of modules or complexes" href="___External__Product.html">ExternalProduct</a> -- external product of modules or complexes</span></li> <li><span><kbd>factorBFunction</kbd> -- see <span><a title="factorization of a b-function" href="_factor__B__Function_lp__Ring__Element_rp.html">factorBFunction(RingElement)</a> -- factorization of a b-function</span></span></li> <li><span><kbd>generalB</kbd> -- see <span><a title="global generalized Bernstein-Sato polynomial" href="_general__B_lp__List_cm__Ring__Element_rp.html">generalB(List,RingElement)</a> -- global generalized Bernstein-Sato polynomial</span></span></li> <li><span><kbd>getIntRoots</kbd> -- see <span><a title="get integer roots of a b-function" href="_get__Int__Roots_lp__Ring__Element_rp.html">getIntRoots(RingElement)</a> -- get integer roots of a b-function</span></span></li> <li><span><kbd>globalB</kbd> -- see <span><a title="compute global b-function and b-operator for a D-module and a polynomial" href="_global__B_lp__Ideal_cm__Ring__Element_rp.html">globalB(Ideal,RingElement)</a> -- compute global b-function and b-operator for a D-module and a polynomial</span></span></li> <li><span><kbd>globalBFunction</kbd> -- see <span><a title="global b-function (else known as the Bernstein-Sato polynomial)" href="_global__B__Function_lp__Ring__Element_rp.html">globalBFunction(RingElement)</a> -- global b-function (else known as the Bernstein-Sato polynomial)</span></span></li> <li><span><kbd>globalBoperator</kbd> -- see <span><a title="compute a b-operator of a polynomial" href="_global__Boperator_lp__Ring__Element_rp.html">globalBoperator(RingElement)</a> -- compute a b-operator of a polynomial</span></span></li> <li><span><kbd>hasRationalSing</kbd> -- see <span><a title="check if a complete intersection has at most rational singularities" href="_has__Rational__Sing_lp__List_rp.html">hasRationalSing(List)</a> -- check if a complete intersection has at most rational singularities</span></span></li> <li><span><a title="intersection (co)homology module of an irreducible closed subvariety" href="___I__Hmodule.html">IHmodule</a> -- intersection (co)homology module of an irreducible closed subvariety</span></li> <li><span><a title="intersection cohomology of an irreducible affine variety" href="_intersection__Cohomology.html">intersectionCohomology</a> -- intersection cohomology of an irreducible affine variety</span></li> <li><span><kbd>isInMultiplierIdeal</kbd> -- see <span><a title="multiplier ideal membership test" href="_is__In__Multiplier__Ideal_lp__Ring__Element_cm__Ideal_cm__Q__Q_rp.html">isInMultiplierIdeal(RingElement,Ideal,QQ)</a> -- multiplier ideal membership test</span></span></li> <li><span><kbd>jumpingCoefficients</kbd> -- see <span><a title="jumping coefficients and corresponding multiplier ideals" href="_jumping__Coefficients_lp__Ideal_rp.html">jumpingCoefficients(Ideal)</a> -- jumping coefficients and corresponding multiplier ideals</span></span></li> <li><span><a title="D-annihilator of 1/f for a planar curve" href="_kappa__Ann__F1__Planar__Curve.html">kappaAnnF1PlanarCurve</a> -- D-annihilator of 1/f for a planar curve</span></li> <li><span><a title="k-th order D-annihilator of a power of a polynomial" href="_k__Order__Ann__Fa.html">kOrderAnnFa</a> -- k-th order D-annihilator of a power of a polynomial</span></li> <li><span><kbd>kOrderAnnFs</kbd> -- see <span><a title="k-th order D-annihilator of a power of a polynomial" href="_k__Order__Ann__Fa.html">kOrderAnnFa</a> -- k-th order D-annihilator of a power of a polynomial</span></span></li> <li><span><kbd>lct</kbd> -- see <span><a title="compute the log canonical threshold for an ideal" href="_lct_lp__Ideal_rp.html">lct(Ideal)</a> -- compute the log canonical threshold for an ideal</span></span></li> <li><span><kbd>localBFunction</kbd> -- see <span><a title="local b-function (a.k.a. the local Bernstein-Sato polynomial)" href="_local__B__Function_lp__Ring__Element_cm__Ideal_rp.html">localBFunction(RingElement,Ideal)</a> -- local b-function (a.k.a. the local Bernstein-Sato polynomial)</span></span></li> <li><span><a title="local cohomology" href="_local__Cohom.html">localCohom</a> -- local cohomology</span></li> <li><span><a title="the characteristic cycle of the localized $D$-module" href="_localize__Characteristic__Cycle.html">localizeCharacteristicCycle</a> -- the characteristic cycle of the localized $D$-module</span></li> <li><span><kbd>logCohomology</kbd> -- see <span><a title="logarithmic cohomology groups in two variables" href="_log__Cohomology_lp__Ring__Element_rp.html">logCohomology(RingElement)</a> -- logarithmic cohomology groups in two variables</span></span></li> <li><span><kbd>multiplierIdeal</kbd> -- see <span><a title="multiplier ideal" href="_multiplier__Ideal_lp__Ideal_cm__Q__Q_rp.html">multiplierIdeal(Ideal,QQ)</a> -- multiplier ideal</span></span></li> <li><span><kbd>paramBpoly</kbd> -- see <span><a title="compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial" href="_param__Bpoly_lp__Ring__Element_cm__File_rp.html">paramBpoly(RingElement,File)</a> -- compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial</span></span></li> <li><span><a title="annihilator of a polynomial in the Weyl algebra" href="_polynomial__Annihilator.html">polynomialAnnihilator</a> -- annihilator of a polynomial in the Weyl algebra</span></li> <li><span><a title="Ext groups between a holonomic module and a polynomial ring" href="_polynomial__Ext.html">polynomialExt</a> -- Ext groups between a holonomic module and a polynomial ring</span></li> <li><span><a title="polynomial solutions of a holonomic system" href="_polynomial__Solutions.html">polynomialSolutions</a> -- polynomial solutions of a holonomic system</span></li> <li><span><kbd>populateCechComplexCC</kbd> -- see <span><a title="Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules" href="_populate__Cech__Complex__C__C_lp__Ideal_cm__List_rp.html">populateCechComplexCC(Ideal,List)</a> -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules</span></span></li> <li><span><kbd>pruneCechComplexCC</kbd> -- see <span><a title="reduction of the Cech complex that produces characteristic cycles of local cohomology modules" href="_prune__Cech__Complex__C__C_lp__Mutable__Hash__Table_rp.html">pruneCechComplexCC(MutableHashTable)</a> -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules</span></span></li> <li><span><kbd>pruneLocalCohom</kbd> -- see <span><a title="prunes local cohomology modules" href="_prune__Local__Cohom_lp__Hash__Table_rp.html">pruneLocalCohom(HashTable)</a> -- prunes local cohomology modules</span></span></li> <li><span><a title="annihilator of a rational function in Weyl algebra" href="_rational__Function__Annihilator.html">rationalFunctionAnnihilator</a> -- annihilator of a rational function in Weyl algebra</span></li> <li><span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></li> <li><span><a title="rational solutions of a holonomic system" href="_rational__Function__Solutions.html">rationalFunctionSolutions</a> -- rational solutions of a holonomic system</span></li> <li><span><a title="Reiffen's curve" href="_reiffen.html">reiffen</a> -- Reiffen's curve</span></li> <li><span><a title="Weyl closure of an ideal" href="___Weyl__Closure.html">WeylClosure</a> -- Weyl closure of an ideal</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>AnnFs(RingElement)</kbd> -- see <span><a title="differential annihilator of a polynomial in a Weyl algebra" href="___Ann__Fs.html">AnnFs</a> -- differential annihilator of a polynomial in a Weyl algebra</span></span></li> <li><span><a title="the annihilating ideal of f^s for an arbitrary D-module" href="___Ann__I__Fs_lp__Ideal_cm__Ring__Element_rp.html">AnnIFs(Ideal,RingElement)</a> -- the annihilating ideal of f^s for an arbitrary D-module</span></li> <li><span><a title="b-function of an ideal" href="_b__Function_lp__Ideal_cm__List_rp.html">bFunction(Ideal,List)</a> -- b-function of an ideal</span></li> <li><span><a title="b-function of a holonomic D-module" href="_b__Function_lp__Module_cm__List_cm__List_rp.html">bFunction(Module,List,List)</a> -- b-function of a holonomic D-module</span></li> <li><span><a title="get roots of a b-function" href="_b__Function__Roots_lp__Ring__Element_rp.html">bFunctionRoots(RingElement)</a> -- get roots of a b-function</span></li> <li><span><kbd>Ddual(Ideal)</kbd> -- see <span><a title="holonomic dual of a D-module" href="___Ddual.html">Ddual</a> -- holonomic dual of a D-module</span></span></li> <li><span><kbd>Ddual(Module)</kbd> -- see <span><a title="holonomic dual of a D-module" href="___Ddual.html">Ddual</a> -- holonomic dual of a D-module</span></span></li> <li><span><kbd>deRham(RingElement)</kbd> -- see <span><a title="deRham cohomology groups for the complement of a hypersurface" href="_de__Rham.html">deRham</a> -- deRham cohomology groups for the complement of a hypersurface</span></span></li> <li><span><kbd>deRham(ZZ,RingElement)</kbd> -- see <span><a title="deRham cohomology groups for the complement of a hypersurface" href="_de__Rham.html">deRham</a> -- deRham cohomology groups for the complement of a hypersurface</span></span></li> <li><span><span class="tt">deRham(Ideal)</span> (missing documentation)<!--tag: (deRham,Ideal)--> </span></li> <li><span><span class="tt">deRham(ZZ,Ideal)</span> (missing documentation)<!--tag: (deRham,ZZ,Ideal)--> </span></li> <li><span><a title="deRham complex for the complement of a hypersurface" href="_de__Rham__All_lp__Ring__Element_rp.html">deRhamAll(RingElement)</a> -- deRham complex for the complement of a hypersurface</span></li> <li><span><kbd>DExt(Module,Module)</kbd> -- see <span><a title="Ext groups between holonomic modules" href="___D__Ext.html">DExt</a> -- Ext groups between holonomic modules</span></span></li> <li><span><kbd>DExt(Module,Module,List)</kbd> -- see <span><a title="Ext groups between holonomic modules" href="___D__Ext.html">DExt</a> -- Ext groups between holonomic modules</span></span></li> <li><span><kbd>DHom(Ideal,Ideal)</kbd> -- see <span><a title="D-homomorphisms between holonomic D-modules" href="___D__Hom.html">DHom</a> -- D-homomorphisms between holonomic D-modules</span></span></li> <li><span><kbd>DHom(Module,Module)</kbd> -- see <span><a title="D-homomorphisms between holonomic D-modules" href="___D__Hom.html">DHom</a> -- D-homomorphisms between holonomic D-modules</span></span></li> <li><span><kbd>DHom(Module,Module,List)</kbd> -- see <span><a title="D-homomorphisms between holonomic D-modules" href="___D__Hom.html">DHom</a> -- D-homomorphisms between holonomic D-modules</span></span></li> <li><span><kbd>diffRatFun(List,RingElement)</kbd> -- see <span><a title="derivative of a rational function in a Weyl algebra" href="_diff__Rat__Fun.html">diffRatFun</a> -- derivative of a rational function in a Weyl algebra</span></span></li> <li><span><kbd>diffRatFun(List,RingElement,RingElement,ZZ)</kbd> -- see <span><a title="derivative of a rational function in a Weyl algebra" href="_diff__Rat__Fun.html">diffRatFun</a> -- derivative of a rational function in a Weyl algebra</span></span></li> <li><span><kbd>Dintegration(Ideal,List)</kbd> -- see <span><a title="integration modules of a D-module" href="___Dintegration.html">Dintegration</a> -- integration modules of a D-module</span></span></li> <li><span><kbd>Dintegration(Module,List)</kbd> -- see <span><a title="integration modules of a D-module" href="___Dintegration.html">Dintegration</a> -- integration modules of a D-module</span></span></li> <li><span><kbd>Dintegration(ZZ,Ideal,List)</kbd> -- see <span><a title="integration modules of a D-module" href="___Dintegration.html">Dintegration</a> -- integration modules of a D-module</span></span></li> <li><span><kbd>Dintegration(ZZ,Module,List)</kbd> -- see <span><a title="integration modules of a D-module" href="___Dintegration.html">Dintegration</a> -- integration modules of a D-module</span></span></li> <li><span><kbd>DintegrationAll(Ideal,List)</kbd> -- see <span><a title="integration modules of a D-module (extended version)" href="___Dintegration__All.html">DintegrationAll</a> -- integration modules of a D-module (extended version)</span></span></li> <li><span><kbd>DintegrationAll(Module,List)</kbd> -- see <span><a title="integration modules of a D-module (extended version)" href="___Dintegration__All.html">DintegrationAll</a> -- integration modules of a D-module (extended version)</span></span></li> <li><span><kbd>DintegrationClasses(Ideal,List)</kbd> -- see <span><a title="integration classes of a D-module" href="___Dintegration__Classes.html">DintegrationClasses</a> -- integration classes of a D-module</span></span></li> <li><span><kbd>DintegrationClasses(Module,List)</kbd> -- see <span><a title="integration classes of a D-module" href="___Dintegration__Classes.html">DintegrationClasses</a> -- integration classes of a D-module</span></span></li> <li><span><kbd>DintegrationClasses(ZZ,Ideal,List)</kbd> -- see <span><a title="integration classes of a D-module" href="___Dintegration__Classes.html">DintegrationClasses</a> -- integration classes of a D-module</span></span></li> <li><span><kbd>DintegrationClasses(ZZ,Module,List)</kbd> -- see <span><a title="integration classes of a D-module" href="___Dintegration__Classes.html">DintegrationClasses</a> -- integration classes of a D-module</span></span></li> <li><span><kbd>DintegrationComplex(Ideal,List)</kbd> -- see <span><a title="derived integration complex of a D-module" href="___Dintegration__Complex.html">DintegrationComplex</a> -- derived integration complex of a D-module</span></span></li> <li><span><kbd>DintegrationComplex(Module,List)</kbd> -- see <span><a title="derived integration complex of a D-module" href="___Dintegration__Complex.html">DintegrationComplex</a> -- derived integration complex of a D-module</span></span></li> <li><span><kbd>DintegrationIdeal(Ideal,List)</kbd> -- see <span><a title="integration ideal of a D-module" href="___Dintegration__Ideal.html">DintegrationIdeal</a> -- integration ideal of a D-module</span></span></li> <li><span><kbd>Dlocalize(Ideal,RingElement)</kbd> -- see <span><a title="localization of a D-module" href="___Dlocalize.html">Dlocalize</a> -- localization of a D-module</span></span></li> <li><span><kbd>Dlocalize(Module,RingElement)</kbd> -- see <span><a title="localization of a D-module" href="___Dlocalize.html">Dlocalize</a> -- localization of a D-module</span></span></li> <li><span><kbd>DlocalizeAll(Ideal,RingElement)</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><kbd>DlocalizeAll(Module,RingElement)</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><kbd>DlocalizeMap(Ideal,RingElement)</kbd> -- see <span><a title="localization map from a D-module to its localization" href="___Dlocalize__Map.html">DlocalizeMap</a> -- localization map from a D-module to its localization</span></span></li> <li><span><kbd>DlocalizeMap(Module,RingElement)</kbd> -- see <span><a title="localization map from a D-module to its localization" href="___Dlocalize__Map.html">DlocalizeMap</a> -- localization map from a D-module to its localization</span></span></li> <li><span><kbd>Dresolution(Ideal)</kbd> -- see <span><a title="resolution of a D-module" href="___Dresolution.html">Dresolution</a> -- resolution of a D-module</span></span></li> <li><span><kbd>Dresolution(Ideal,List)</kbd> -- see <span><a title="resolution of a D-module" href="___Dresolution.html">Dresolution</a> -- resolution of a D-module</span></span></li> <li><span><kbd>Dresolution(Module)</kbd> -- see <span><a title="resolution of a D-module" href="___Dresolution.html">Dresolution</a> -- resolution of a D-module</span></span></li> <li><span><kbd>Dresolution(Module,List)</kbd> -- see <span><a title="resolution of a D-module" href="___Dresolution.html">Dresolution</a> -- resolution of a D-module</span></span></li> <li><span><kbd>Drestriction(Ideal,List)</kbd> -- see <span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></span></li> <li><span><kbd>Drestriction(Module,List)</kbd> -- see <span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></span></li> <li><span><kbd>Drestriction(ZZ,Ideal,List)</kbd> -- see <span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></span></li> <li><span><kbd>Drestriction(ZZ,Module,List)</kbd> -- see <span><a title="restriction modules of a D-module" href="___Drestriction.html">Drestriction</a> -- restriction modules of a D-module</span></span></li> <li><span><kbd>DrestrictionAll(Ideal,List)</kbd> -- see <span><a title="restriction modules of a D-module (extended version)" href="___Drestriction__All.html">DrestrictionAll</a> -- restriction modules of a D-module (extended version)</span></span></li> <li><span><kbd>DrestrictionAll(Module,List)</kbd> -- see <span><a title="restriction modules of a D-module (extended version)" href="___Drestriction__All.html">DrestrictionAll</a> -- restriction modules of a D-module (extended version)</span></span></li> <li><span><kbd>DrestrictionClasses(Ideal,List)</kbd> -- see <span><a title="restriction classes of a D-module" href="___Drestriction__Classes.html">DrestrictionClasses</a> -- restriction classes of a D-module</span></span></li> <li><span><kbd>DrestrictionClasses(Module,List)</kbd> -- see <span><a title="restriction classes of a D-module" href="___Drestriction__Classes.html">DrestrictionClasses</a> -- restriction classes of a D-module</span></span></li> <li><span><kbd>DrestrictionClasses(ZZ,Ideal,List)</kbd> -- see <span><a title="restriction classes of a D-module" href="___Drestriction__Classes.html">DrestrictionClasses</a> -- restriction classes of a D-module</span></span></li> <li><span><kbd>DrestrictionClasses(ZZ,Module,List)</kbd> -- see <span><a title="restriction classes of a D-module" href="___Drestriction__Classes.html">DrestrictionClasses</a> -- restriction classes of a D-module</span></span></li> <li><span><kbd>DrestrictionComplex(Ideal,List)</kbd> -- see <span><a title="derived restriction complex of a D-module" href="___Drestriction__Complex.html">DrestrictionComplex</a> -- derived restriction complex of a D-module</span></span></li> <li><span><kbd>DrestrictionComplex(Module,List)</kbd> -- see <span><a title="derived restriction complex of a D-module" href="___Drestriction__Complex.html">DrestrictionComplex</a> -- derived restriction complex of a D-module</span></span></li> <li><span><kbd>DrestrictionIdeal(Ideal,List)</kbd> -- see <span><a title="restriction ideal of a D-module" href="___Drestriction__Ideal.html">DrestrictionIdeal</a> -- restriction ideal of a D-module</span></span></li> <li><span><kbd>ExternalProduct(ChainComplex,ChainComplex)</kbd> -- see <span><a title="external product of modules or complexes" href="___External__Product.html">ExternalProduct</a> -- external product of modules or complexes</span></span></li> <li><span><kbd>ExternalProduct(Module,Module)</kbd> -- see <span><a title="external product of modules or complexes" href="___External__Product.html">ExternalProduct</a> -- external product of modules or complexes</span></span></li> <li><span><a title="factorization of a b-function" href="_factor__B__Function_lp__Ring__Element_rp.html">factorBFunction(RingElement)</a> -- factorization of a b-function</span></li> <li><span><kbd>generalB(List)</kbd> -- see <span><a title="global generalized Bernstein-Sato polynomial" href="_general__B_lp__List_cm__Ring__Element_rp.html">generalB(List,RingElement)</a> -- global generalized Bernstein-Sato polynomial</span></span></li> <li><span><a title="global generalized Bernstein-Sato polynomial" href="_general__B_lp__List_cm__Ring__Element_rp.html">generalB(List,RingElement)</a> -- global generalized Bernstein-Sato polynomial</span></li> <li><span><a title="get integer roots of a b-function" href="_get__Int__Roots_lp__Ring__Element_rp.html">getIntRoots(RingElement)</a> -- get integer roots of a b-function</span></li> <li><span><a title="compute global b-function and b-operator for a D-module and a polynomial" href="_global__B_lp__Ideal_cm__Ring__Element_rp.html">globalB(Ideal,RingElement)</a> -- compute global b-function and b-operator for a D-module and a polynomial</span></li> <li><span><a title="global b-function (else known as the Bernstein-Sato polynomial)" href="_global__B__Function_lp__Ring__Element_rp.html">globalBFunction(RingElement)</a> -- global b-function (else known as the Bernstein-Sato polynomial)</span></li> <li><span><a title="compute a b-operator of a polynomial" href="_global__Boperator_lp__Ring__Element_rp.html">globalBoperator(RingElement)</a> -- compute a b-operator of a polynomial</span></li> <li><span><a title="check if a complete intersection has at most rational singularities" href="_has__Rational__Sing_lp__List_rp.html">hasRationalSing(List)</a> -- check if a complete intersection has at most rational singularities</span></li> <li><span><kbd>IHmodule(Ideal)</kbd> -- see <span><a title="intersection (co)homology module of an irreducible closed subvariety" href="___I__Hmodule.html">IHmodule</a> -- intersection (co)homology module of an irreducible closed subvariety</span></span></li> <li><span><kbd>intersectionCohomology(Ideal)</kbd> -- see <span><a title="intersection cohomology of an irreducible affine variety" href="_intersection__Cohomology.html">intersectionCohomology</a> -- intersection cohomology of an irreducible affine variety</span></span></li> <li><span><kbd>intersectionCohomology(ZZ,Ideal)</kbd> -- see <span><a title="intersection cohomology of an irreducible affine variety" href="_intersection__Cohomology.html">intersectionCohomology</a> -- intersection cohomology of an irreducible affine variety</span></span></li> <li><span><a title="multiplier ideal membership test" href="_is__In__Multiplier__Ideal_lp__Ring__Element_cm__Ideal_cm__Q__Q_rp.html">isInMultiplierIdeal(RingElement,Ideal,QQ)</a> -- multiplier ideal membership test</span></li> <li><span><a title="jumping coefficients and corresponding multiplier ideals" href="_jumping__Coefficients_lp__Ideal_rp.html">jumpingCoefficients(Ideal)</a> -- jumping coefficients and corresponding multiplier ideals</span></li> <li><span><kbd>jumpingCoefficients(Ideal,QQ,QQ)</kbd> -- see <span><a title="jumping coefficients and corresponding multiplier ideals" href="_jumping__Coefficients_lp__Ideal_rp.html">jumpingCoefficients(Ideal)</a> -- jumping coefficients and corresponding multiplier ideals</span></span></li> <li><span><kbd>jumpingCoefficients(Ideal,QQ,ZZ)</kbd> -- see <span><a title="jumping coefficients and corresponding multiplier ideals" href="_jumping__Coefficients_lp__Ideal_rp.html">jumpingCoefficients(Ideal)</a> -- jumping coefficients and corresponding multiplier ideals</span></span></li> <li><span><kbd>jumpingCoefficients(Ideal,ZZ,QQ)</kbd> -- see <span><a title="jumping coefficients and corresponding multiplier ideals" href="_jumping__Coefficients_lp__Ideal_rp.html">jumpingCoefficients(Ideal)</a> -- jumping coefficients and corresponding multiplier ideals</span></span></li> <li><span><kbd>jumpingCoefficients(Ideal,ZZ,ZZ)</kbd> -- see <span><a title="jumping coefficients and corresponding multiplier ideals" href="_jumping__Coefficients_lp__Ideal_rp.html">jumpingCoefficients(Ideal)</a> -- jumping coefficients and corresponding multiplier ideals</span></span></li> <li><span><kbd>kappaAnnF1PlanarCurve(RingElement)</kbd> -- see <span><a title="D-annihilator of 1/f for a planar curve" href="_kappa__Ann__F1__Planar__Curve.html">kappaAnnF1PlanarCurve</a> -- D-annihilator of 1/f for a planar curve</span></span></li> <li><span><kbd>kOrderAnnFa(ZZ,RingElement,ZZ)</kbd> -- see <span><a title="k-th order D-annihilator of a power of a polynomial" href="_k__Order__Ann__Fa.html">kOrderAnnFa</a> -- k-th order D-annihilator of a power of a polynomial</span></span></li> <li><span><kbd>kOrderAnnFs(ZZ,RingElement)</kbd> -- see <span><a title="k-th order D-annihilator of a power of a polynomial" href="_k__Order__Ann__Fa.html">kOrderAnnFa</a> -- k-th order D-annihilator of a power of a polynomial</span></span></li> <li><span><a title="compute the log canonical threshold for an ideal" href="_lct_lp__Ideal_rp.html">lct(Ideal)</a> -- compute the log canonical threshold for an ideal</span></li> <li><span><a title="local b-function (a.k.a. the local Bernstein-Sato polynomial)" href="_local__B__Function_lp__Ring__Element_cm__Ideal_rp.html">localBFunction(RingElement,Ideal)</a> -- local b-function (a.k.a. the local Bernstein-Sato polynomial)</span></li> <li><span><a title="local cohomology of a polynomial ring" href="_local__Cohom_lp__Ideal_rp.html">localCohom(Ideal)</a> -- local cohomology of a polynomial ring</span></li> <li><span><a title="local cohomology of a D-module" href="_local__Cohom_lp__Ideal_cm__Module_rp.html">localCohom(Ideal,Module)</a> -- local cohomology of a D-module</span></li> <li><span><a title="local cohomology of a polynomial ring" href="_local__Cohom_lp__List_cm__Ideal_rp.html">localCohom(List,Ideal)</a> -- local cohomology of a polynomial ring</span></li> <li><span><a title="local cohomology of a D-module" href="_local__Cohom_lp__List_cm__Ideal_cm__Module_rp.html">localCohom(List,Ideal,Module)</a> -- local cohomology of a D-module</span></li> <li><span><a title="local cohomology of a polynomial ring" href="_local__Cohom_lp__Z__Z_cm__Ideal_rp.html">localCohom(ZZ,Ideal)</a> -- local cohomology of a polynomial ring</span></li> <li><span><a title="local cohomology of a D-module" href="_local__Cohom_lp__Z__Z_cm__Ideal_cm__Module_rp.html">localCohom(ZZ,Ideal,Module)</a> -- local cohomology of a D-module</span></li> <li><span><kbd>localizeCharacteristicCycle(Ideal,RingElement)</kbd> -- see <span><a title="the characteristic cycle of the localized $D$-module" href="_localize__Characteristic__Cycle.html">localizeCharacteristicCycle</a> -- the characteristic cycle of the localized $D$-module</span></span></li> <li><span><kbd>localizeCharacteristicCycle(List,RingElement)</kbd> -- see <span><a title="the characteristic cycle of the localized $D$-module" href="_localize__Characteristic__Cycle.html">localizeCharacteristicCycle</a> -- the characteristic cycle of the localized $D$-module</span></span></li> <li><span><a title="logarithmic cohomology groups in two variables" href="_log__Cohomology_lp__Ring__Element_rp.html">logCohomology(RingElement)</a> -- logarithmic cohomology groups in two variables</span></li> <li><span><kbd>multiplierIdeal(Ideal,List)</kbd> -- see <span><a title="multiplier ideal" href="_multiplier__Ideal_lp__Ideal_cm__Q__Q_rp.html">multiplierIdeal(Ideal,QQ)</a> -- multiplier ideal</span></span></li> <li><span><a title="multiplier ideal" href="_multiplier__Ideal_lp__Ideal_cm__Q__Q_rp.html">multiplierIdeal(Ideal,QQ)</a> -- multiplier ideal</span></li> <li><span><kbd>multiplierIdeal(Ideal,ZZ)</kbd> -- see <span><a title="multiplier ideal" href="_multiplier__Ideal_lp__Ideal_cm__Q__Q_rp.html">multiplierIdeal(Ideal,QQ)</a> -- multiplier ideal</span></span></li> <li><span><a title="compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial" href="_param__Bpoly_lp__Ring__Element_cm__File_rp.html">paramBpoly(RingElement,File)</a> -- compute the list of all possible Bernstein-Sato polynomials for a parametric polynomial</span></li> <li><span><kbd>polynomialAnnihilator(RingElement)</kbd> -- see <span><a title="annihilator of a polynomial in the Weyl algebra" href="_polynomial__Annihilator.html">polynomialAnnihilator</a> -- annihilator of a polynomial in the Weyl algebra</span></span></li> <li><span><kbd>polynomialExt(Ideal)</kbd> -- see <span><a title="Ext groups between a holonomic module and a polynomial ring" href="_polynomial__Ext.html">polynomialExt</a> -- Ext groups between a holonomic module and a polynomial ring</span></span></li> <li><span><kbd>polynomialExt(Module)</kbd> -- see <span><a title="Ext groups between a holonomic module and a polynomial ring" href="_polynomial__Ext.html">polynomialExt</a> -- Ext groups between a holonomic module and a polynomial ring</span></span></li> <li><span><kbd>polynomialExt(ZZ,Ideal)</kbd> -- see <span><a title="Ext groups between a holonomic module and a polynomial ring" href="_polynomial__Ext.html">polynomialExt</a> -- Ext groups between a holonomic module and a polynomial ring</span></span></li> <li><span><kbd>polynomialExt(ZZ,Module)</kbd> -- see <span><a title="Ext groups between a holonomic module and a polynomial ring" href="_polynomial__Ext.html">polynomialExt</a> -- Ext groups between a holonomic module and a polynomial ring</span></span></li> <li><span><kbd>polynomialSolutions(Ideal)</kbd> -- see <span><a title="polynomial solutions of a holonomic system" href="_polynomial__Solutions.html">polynomialSolutions</a> -- polynomial solutions of a holonomic system</span></span></li> <li><span><kbd>polynomialSolutions(Ideal,List)</kbd> -- see <span><a title="polynomial solutions of a holonomic system" href="_polynomial__Solutions.html">polynomialSolutions</a> -- polynomial solutions of a holonomic system</span></span></li> <li><span><kbd>polynomialSolutions(Module)</kbd> -- see <span><a title="polynomial solutions of a holonomic system" href="_polynomial__Solutions.html">polynomialSolutions</a> -- polynomial solutions of a holonomic system</span></span></li> <li><span><kbd>polynomialSolutions(Module,List)</kbd> -- see <span><a title="polynomial solutions of a holonomic system" href="_polynomial__Solutions.html">polynomialSolutions</a> -- polynomial solutions of a holonomic system</span></span></li> <li><span><a title="Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules" href="_populate__Cech__Complex__C__C_lp__Ideal_cm__List_rp.html">populateCechComplexCC(Ideal,List)</a> -- Cech complex skeleton for the computation of the characteristic cycles of local cohomology modules</span></li> <li><span><a title="reduction of the Cech complex that produces characteristic cycles of local cohomology modules" href="_prune__Cech__Complex__C__C_lp__Mutable__Hash__Table_rp.html">pruneCechComplexCC(MutableHashTable)</a> -- reduction of the Cech complex that produces characteristic cycles of local cohomology modules</span></li> <li><span><a title="prunes local cohomology modules" href="_prune__Local__Cohom_lp__Hash__Table_rp.html">pruneLocalCohom(HashTable)</a> -- prunes local cohomology modules</span></li> <li><span><kbd>rationalFunctionAnnihilator(RingElement)</kbd> -- see <span><a title="annihilator of a rational function in Weyl algebra" href="_rational__Function__Annihilator.html">rationalFunctionAnnihilator</a> -- annihilator of a rational function in Weyl algebra</span></span></li> <li><span><kbd>rationalFunctionAnnihilator(RingElement,RingElement)</kbd> -- see <span><a title="annihilator of a rational function in Weyl algebra" href="_rational__Function__Annihilator.html">rationalFunctionAnnihilator</a> -- annihilator of a rational function in Weyl algebra</span></span></li> <li><span><kbd>rationalFunctionExt(Ideal)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionExt(Ideal,RingElement)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionExt(Module)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionExt(Module,RingElement)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionExt(ZZ,Ideal)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionExt(ZZ,Ideal,RingElement)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionExt(ZZ,Module)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionExt(ZZ,Module,RingElement)</kbd> -- see <span><a title="Ext(holonomic D-module, polynomial ring localized at the singular locus)" href="_rational__Function__Ext.html">rationalFunctionExt</a> -- Ext(holonomic D-module, polynomial ring localized at the singular locus)</span></span></li> <li><span><kbd>rationalFunctionSolutions(Ideal)</kbd> -- see <span><a title="rational solutions of a holonomic system" href="_rational__Function__Solutions.html">rationalFunctionSolutions</a> -- rational solutions of a holonomic system</span></span></li> <li><span><kbd>rationalFunctionSolutions(Ideal,List)</kbd> -- see <span><a title="rational solutions of a holonomic system" href="_rational__Function__Solutions.html">rationalFunctionSolutions</a> -- rational solutions of a holonomic system</span></span></li> <li><span><kbd>rationalFunctionSolutions(Ideal,List,List)</kbd> -- see <span><a title="rational solutions of a holonomic system" href="_rational__Function__Solutions.html">rationalFunctionSolutions</a> -- rational solutions of a holonomic system</span></span></li> <li><span><kbd>rationalFunctionSolutions(Ideal,RingElement)</kbd> -- see <span><a title="rational solutions of a holonomic system" href="_rational__Function__Solutions.html">rationalFunctionSolutions</a> -- rational solutions of a holonomic system</span></span></li> <li><span><kbd>rationalFunctionSolutions(Ideal,RingElement,List)</kbd> -- see <span><a title="rational solutions of a holonomic system" href="_rational__Function__Solutions.html">rationalFunctionSolutions</a> -- rational solutions of a holonomic system</span></span></li> <li><span><kbd>reiffen(ZZ,ZZ)</kbd> -- see <span><a title="Reiffen's curve" href="_reiffen.html">reiffen</a> -- Reiffen's curve</span></span></li> <li><span><kbd>WeylClosure(Ideal)</kbd> -- see <span><a title="Weyl closure of an ideal" href="___Weyl__Closure.html">WeylClosure</a> -- Weyl closure of an ideal</span></span></li> <li><span><kbd>WeylClosure(Ideal,RingElement)</kbd> -- see <span><a title="Weyl closure of an ideal" href="___Weyl__Closure.html">WeylClosure</a> -- Weyl closure of an ideal</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><a href="___Alg.html">Alg</a></span></li> <li><span><a title="a key in the hashtable created by Drestriction/Dintegration" href="___B__Function.html">BFunction</a> -- a key in the hashtable created by Drestriction/Dintegration</span></li> <li><span><kbd>IntRing</kbd> -- see <span><a title="specify strategy for computing b-function" href="_b__Function_lp..._cm__Strategy_eq_gt..._rp.html">bFunction(...,Strategy=>...)</a> -- specify strategy for computing b-function</span></span></li> <li><span><kbd>NonGeneric</kbd> -- see <span><a title="specify strategy for computing b-function" href="_b__Function_lp..._cm__Strategy_eq_gt..._rp.html">bFunction(...,Strategy=>...)</a> -- specify strategy for computing b-function</span></span></li> <li><span><kbd>TryGeneric</kbd> -- see <span><a title="specify strategy for computing b-function" href="_b__Function_lp..._cm__Strategy_eq_gt..._rp.html">bFunction(...,Strategy=>...)</a> -- specify strategy for computing b-function</span></span></li> <li><span><a title="a key attached by globalB and Dlocalize" href="___Boperator.html">Boperator</a> -- a key attached by globalB and Dlocalize</span></li> <li><span><a title="a key in the hashtable created by Drestriction/Dintegration" href="___Boundaries.html">Boundaries</a> -- a key in the hashtable created by Drestriction/Dintegration</span></li> <li><span><a title="a key attached by globalB" href="___Bpolynomial.html">Bpolynomial</a> -- a key attached by globalB</span></li> <li><span><a title="a key in the hashtable created by deRham" href="___Cohomology__Groups.html">CohomologyGroups</a> -- a key in the hashtable created by deRham</span></li> <li><span><a title="a key in the hashtable created by Drestriction/Dintegration" href="___Cycles.html">Cycles</a> -- a key in the hashtable created by Drestriction/Dintegration</span></li> <li><span><kbd>Oaku</kbd> -- see <span><a title="strategy for computing a localization of a D-module" href="___Dlocalize_lp..._cm__Strategy_eq_gt..._rp.html">Dlocalize(...,Strategy=>...)</a> -- strategy for computing a localization of a D-module</span></span></li> <li><span><kbd>OTW</kbd> -- see <span><a title="strategy for computing a localization of a D-module" href="___Dlocalize_lp..._cm__Strategy_eq_gt..._rp.html">Dlocalize(...,Strategy=>...)</a> -- strategy for computing a localization of a D-module</span></span></li> <li><span><kbd>OTWcyclic</kbd> -- see <span><a title="strategy for computing a localization of a D-module" href="___Dlocalize_lp..._cm__Strategy_eq_gt..._rp.html">Dlocalize(...,Strategy=>...)</a> -- strategy for computing a localization of a D-module</span></span></li> <li><span><kbd>annFS</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><kbd>Bfunction</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><kbd>GeneratorPower</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><kbd>IntegrateBfunction</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><kbd>LocMap</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><kbd>LocModule</kbd> -- see <span><a title="localization of a D-module (extended version)" href="___Dlocalize__All.html">DlocalizeAll</a> -- localization of a D-module (extended version)</span></span></li> <li><span><a title="an option for polynomialSolutions=>Alg" href="___Duality.html">Duality</a> -- an option for polynomialSolutions=>Alg</span></li> <li><span><a title="a key in the hashtable created by Drestriction/Dintegration" href="___Explicit.html">Explicit</a> -- a key in the hashtable created by Drestriction/Dintegration</span></li> <li><span><a title="a key in the hashtable created by Drestriction/Dintegration" href="___Exponents.html">Exponents</a> -- a key in the hashtable created by Drestriction/Dintegration</span></li> <li><span><a title="an option for polynomialSolutions=>Alg" href="___G__D.html">GD</a> -- an option for polynomialSolutions=>Alg</span></li> <li><span><a title="a key in the hashtable created by Drestriction/Dintegration" href="___Gen__Cycles.html">GenCycles</a> -- a key in the hashtable created by Drestriction/Dintegration</span></li> <li><span><kbd>Exponent</kbd> -- see <span><a title="specify exponent m for m-generalized Bernstein-Sato polynomial" href="_general__B_lp..._cm__Exponent_eq_gt..._rp.html">generalB(...,Exponent=>...)</a> -- specify exponent m for m-generalized Bernstein-Sato polynomial</span></span></li> <li><span><kbd>InitialIdeal</kbd> -- see <span><a title="specify strategy for computing generalized Bernstein-Sato polynomial" href="_general__B_lp..._cm__Strategy_eq_gt..._rp.html">generalB(...,Strategy=>...)</a> -- specify strategy for computing generalized Bernstein-Sato polynomial</span></span></li> <li><span><kbd>StarIdeal</kbd> -- see <span><a title="specify strategy for computing generalized Bernstein-Sato polynomial" href="_general__B_lp..._cm__Strategy_eq_gt..._rp.html">generalB(...,Strategy=>...)</a> -- specify strategy for computing generalized Bernstein-Sato polynomial</span></span></li> <li><span><a title="a strategy option for lct, globalBFunction" href="___General__Bernstein__Sato.html">GeneralBernsteinSato</a> -- a strategy option for lct, globalBFunction</span></li> <li><span><kbd>ReducedB</kbd> -- see <span><a title="specify strategy for computing global b-function" href="_global__B__Function_lp..._cm__Strategy_eq_gt..._rp.html">globalBFunction(...,Strategy=>...)</a> -- specify strategy for computing global b-function</span></span></li> <li><span><kbd>ViaAnnFs</kbd> -- see <span><a title="specify strategy for computing global b-function" href="_global__B__Function_lp..._cm__Strategy_eq_gt..._rp.html">globalBFunction(...,Strategy=>...)</a> -- specify strategy for computing global b-function</span></span></li> <li><span><a title="a key in a hashtable; an option of DExt" href="___Homology__Modules.html">HomologyModules</a> -- a key in a hashtable; an option of DExt</span></li> <li><span><kbd>LocCohomStrategy</kbd> -- see <span><a title="intersection (co)homology module of an irreducible closed subvariety" href="___I__Hmodule.html">IHmodule</a> -- intersection (co)homology module of an irreducible closed subvariety</span></span></li> <li><span><a href="___Info.html">Info</a></span></li> <li><span><a title="a key in the hashtable created by Dintegration" href="___Integrate__Complex.html">IntegrateComplex</a> -- a key in the hashtable created by Dintegration</span></li> <li><span><kbd>generalizedBFunction</kbd> -- see <span><a title="multiplier ideal membership test" href="_is__In__Multiplier__Ideal_lp__Ring__Element_cm__Ideal_cm__Q__Q_rp.html">isInMultiplierIdeal(RingElement,Ideal,QQ)</a> -- multiplier ideal membership test</span></span></li> <li><span><kbd>mGeneralizedBFunction</kbd> -- see <span><a title="multiplier ideal membership test" href="_is__In__Multiplier__Ideal_lp__Ring__Element_cm__Ideal_cm__Q__Q_rp.html">isInMultiplierIdeal(RingElement,Ideal,QQ)</a> -- multiplier ideal membership test</span></span></li> <li><span><a title="a key in the hashtable created by deRham" href="___Localize__Map.html">LocalizeMap</a> -- a key in the hashtable created by deRham</span></li> <li><span><a href="___Loc__Strategy.html">LocStrategy</a></span></li> <li><span><a title="an option for DExt=>Special" href="___None.html">None</a> -- an option for DExt=>Special</span></li> <li><span><a title="an option for localCohom=>Strategy" href="___Oa__Ta.html">OaTa</a> -- an option for localCohom=>Strategy</span></li> <li><span><a title="an option for localCohom => LocStrategy" href="___Oa__Ta__Wa.html">OaTaWa</a> -- an option for localCohom => LocStrategy</span></li> <li><span><a title="a key in the hashtable created by deRham" href="___Omega__Res.html">OmegaRes</a> -- a key in the hashtable created by deRham</span></li> <li><span><a href="___Output.html">Output</a></span></li> <li><span><a title="a key in the hashtable created by deRham" href="___Pre__Cycles.html">PreCycles</a> -- a key in the hashtable created by deRham</span></li> <li><span><a title="a key attached by ExternalProduct" href="_proj__Map1.html">projMap1</a> -- a key attached by ExternalProduct</span></li> <li><span><a title="a key attached by ExternalProduct" href="_proj__Map2.html">projMap2</a> -- a key attached by ExternalProduct</span></li> <li><span><a title="strategy for computing a resolution of a D-module" href="___Schreyer.html">Schreyer</a> -- strategy for computing a resolution of a D-module</span></li> <li><span><a href="___Special.html">Special</a></span></li> <li><span><a title="a key in the hashtable created by deRham" href="___Transfer__Cycles.html">TransferCycles</a> -- a key in the hashtable created by deRham</span></li> <li><span><a title="a key attached by ExternalProduct" href="_twist__Inv__Map.html">twistInvMap</a> -- a key attached by ExternalProduct</span></li> <li><span><a title="indicates whether TwistMap should be computed" href="___Twist__Map.html">TwistMap</a> -- indicates whether TwistMap should be computed</span></li> <li><span><a title="a key attached by ExternalProduct" href="_twist__Map.html">twistMap</a> -- a key attached by ExternalProduct</span></li> <li><span><a title="strategy for computing a resolution of a D-module" href="___Vhomogenize.html">Vhomogenize</a> -- strategy for computing a resolution of a D-module</span></li> <li><span><a title="a strategy option for lct" href="___Via__B__Function.html">ViaBFunction</a> -- a strategy option for lct</span></li> <li><span><span class="tt">ViaColonIdeal</span> (missing documentation)<!--tag: ViaColonIdeal--> </span></li> <li><span><span class="tt">ViaElimination</span> (missing documentation)<!--tag: ViaElimination--> </span></li> <li><span><a title="an option for generalB=>Strategy" href="___Via__Linear__Algebra.html">ViaLinearAlgebra</a> -- an option for generalB=>Strategy</span></li> <li><span><a title="a key in the hashtable created by Drestriction/Dintegration" href="___V__Resolution.html">VResolution</a> -- a key in the hashtable created by Drestriction/Dintegration</span></li> <li><span><a title="an option for localCohom=>Strategy" href="___Walther.html">Walther</a> -- an option for localCohom=>Strategy</span></li> </ul> </li> <li>Other things <ul> <li><span><a title="intersection cohomology of an irreducible affine variety" href="_intersection__Cohomology.html">IH</a> -- intersection cohomology of an irreducible affine variety</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="algorithms for b-functions, local cohomology, and intersection cohomology" href="index.html">BernsteinSato</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">BernsteinSato.m2</span>, with auxiliary files in <span class="tt">BernsteinSato/</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">BernsteinSato/DOC/main.m2:52:0</span>.</p> </div> </div> </div> </body> </html>