Local Bernstein–Sato ideals: Algorithm and examples

AbstractLet 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 non-principal Bernstein–Sato ideals, computed with our algorithm by using the computer algebra system Risa/Asir