Local Bernstein-Sato ideals: Algorithm and examples

Let k be a field of characteristic 0. Given a polynomial mapping f = (f1, . . . , fp) from kn to kp, the local Bernstein–Sato ideal of f at a point a ∈ kn is defined as an ideal of the ring of polynomials in s = (s1, . . . , sp). We propose an algorithm for computing local Bernstein–Sato ideals by combining Gröbner bases in rings of differential operators with primary decomposition in a polynomial ring. It also enables us to compute a constructible stratification of kn such that the local Bernstein–Sato ideal is constant along each stratum.We also present examples, some of which have nonprincipal Bernstein–Sato ideals, computed with our algorithm by using the computer algebra system Risa/Asir. © 2009 Elsevier Ltd. All rights reserved.