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This paper is the widely extended version of the publication, appeared inProceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details onproofs, present new algorithms and examples. We present a general algorithm forcomputing an intersection of a left ideal of an associative algebra over afield with a subalgebra, generated by a single element. We show applications ofthis algorithm in different algebraic situations and describe ourimplementation in \textsc{Singular}. Among other, we use this algorithm incomputational $D$-module theory for computing e. g. the Bernstein-Satopolynomial of a single polynomial with several approaches. We also present anew method, having no analogues yet, for the computation of the Bernstein-Satopolynomial of an affine variety. Also, we provide a new proof of the algorithmby Brian\c{c}on-Maisonobe for the computation of the $s$-parametric annihilatorof a polynomial. Moreover, we present new methods for the latter computation aswell as optimized algorithms for the computation of Bernstein-Sato polynomialin various settings.

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