Algorithms for Checking Rational Roots of $b$-functions and their Applications

TL;DR

This paper introduces algorithms for efficiently checking rational roots of Bernstein-Sato polynomials, enabling new methods for computing these polynomials and their applications in singularity theory and algebraic geometry.

Contribution

It proposes a family of algorithms called checkRoot for optimized root verification and multiplicity computation, improving the computation of Bernstein-Sato polynomials without primary decomposition.

Findings

Algorithms successfully verify rational roots of Bernstein-Sato polynomials.

New approach computes global and local Bernstein-Sato polynomials using bounds from resolutions.

Implementation in Singular libraries demonstrates practical applicability.

Abstract

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called \texttt{checkRoot} for optimized check of whether a given rational number is a root of Bernstein-Sato polynomial and the computations of its multiplicity. This algorithms are used in the new approach to compute the whole global or local Bernstein-Sato polynomial and $b$-function of a holonomic ideal with respect to weights. They are applied in numerous situations, where there is a possibility to compute an upper bound for the polynomial. Namely, it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula of A'Campo and spectral numbers. We also present approaches to the logarithmic comparison problem and the intersection homology D-module. Several applications are presented as well as solutions to some challenges which were intractable with the classical methods. One of the main applications consists of computing of a stratification of affine space with the local $b$-function being constant on each stratum. Notably, the algorithm we propose does not employ primary decomposition. Also we apply our results for the computation of Bernstein-Sato polynomials for varieties. The methods from this paper have been implemented in {\sc Singular:Plural} as libraries {\tt dmod.lib} and {\tt bfun.lib}. All the examples from the paper have been computed with this implementation.