Algorithms for checking rational roots of b-functions and their applications - Levandovskyy, V.; Martín-Morales, J. (Journal of algebra, 2012)
The Bernstein–Sato polynomial of a hypersurface is an important object with many applications. However, its computation is hard, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called checkRoot for optimized checking whether a given rational number is a root of Bernstein–Sato polynomial and in the affirmative case, computing its multiplicity. These algorithms are used in the new approach to compute the global or local Bernstein–Sato polynomial and b-function of a holonomic ideal with respect to a weight vector. They can be applied in numerous situations, where a multiple of the Bernstein–Sato polynomial can be established. Namely, a multiple can be obtained by means of embedded resolution, for topologically equivalent singularities or using the formula of AʼCampo 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 is the computation 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. Our results can be also applied for the computation of Bernstein–Sato polynomials for varieties. The examples in the paper have been computed with our implementation of the methods described in Singular:Plural.