2010/03/17 by Viktor Levandovskyy, Levandovskyy, Viktor, Jorge Martín-Morales +1
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.1003.3478
openalex publication_date 2010/03/17 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
Bernstein-Sato polynomial of a hypersurface is an important object with\nnumerous applications. It is known, that it is complicated to obtain it\ncomputationally, as a number of open questions and challenges indicate. In this\npaper we propose a family of algorithms called \checkRoot for optimized\ncheck of whether a given rational number is a root of Bernstein-Sato polynomial\nand the computations of its multiplicity. This algorithms are used in the new\napproach to compute the whole global or local Bernstein-Sato polynomial and\nb-function of a holonomic ideal with respect to weights. They are applied in\nnumerous situations, where there is a possibility to compute an upper bound for\nthe polynomial. Namely, it can be achieved by means of embedded resolution, for\ntopologically equivalent singularities or using the formula of A'Campo and\nspectral numbers. We also present approaches to the logarithmic comparison\nproblem and the intersection homology D-module. Several applications are\npresented as well as solutions to some challenges which were intractable with\nthe classical methods. One of the main applications consists of computing of a\nstratification of affine space with the local b-function being constant on\neach stratum. Notably, the algorithm we propose does not employ primary\ndecomposition. Also we apply our results for the computation of Bernstein-Sato\npolynomials for varieties. The methods from this paper have been implemented in\n sc Singular:Plural as libraries tt dmod.lib and tt bfun.lib. All the\nexamples from the paper have been computed with this implementation.\n