Effective Methods for the Computation of Bernstein-Sato polynomials for Hypersurfaces and Affine Varieties

This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC’2009 conference (Andres, Levandovskyy, and Mart́ın-Morales, 2009). We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element. We show applications of this algorithm in different algebraic situations and describe our implementation in Singular. Among other, we use this algorithm in computational D-module theory for computing e. g. the Bernstein-Sato polynomial of a single polynomial with several approaches. We also present a new method, having no analogues yet, for the computation of the Bernstein-Sato polynomial of an affine variety. Also, we provide a new proof of the algorithm by Briançon-Maisonobe for the computation of the s-parametric annihilator of a polynomial. Moreover, we present new methods for the latter computation as well as optimized algorithms for the computation of Bernstein-Sato polynomial in various settings.