Definitive Computation of Bernstein-Sato Polynomials

Abstract

Let n and d be positive integers, let k be a field and let P(n,d;k) be the space of the polynomials in n variables of degree at most d with coefficients in k. Let B(n,d) be the set of the Bernstein-Sato polynomials of all polynomials in P(n,d;k) as k varies over all fields of characteristic 0. G. Lyubeznik proved that B(n,d) is a finite set and asked if, for a fixed k, the set of the polynomials corresponding to each element of B(n,d) is a constructible subset of P(n,d;k). In this paper we give an affirmative answer to Lyubeznik's question by showing that the set in question is indeed constructible and defined over Q, i.e. its defining equations are the same for all fields k. Moreover, we construct an algorithm that for each pair (n,d) produces a complete list of the elements of B(n,d) and, for each element of this list, an explicit description of the constructible set of polynomials having this particular Bernstein-Sato polynomial.