Constructibility of the Set of Polynomials with a Fixed Bernstein–Sato Polynomial: an Algorithmic Approach

Letnanddbe positive integers, letkbe a field and letP(n,d ;k)be the space of the non-zero polynomials innvariables of degree at mostdwith coefficients ink. LetB(n, d)be the set of the Bernstein–Sato polynomials of all polynomials inP(n,d ;k)askvaries over all fields of characteristic0. G. Lyubeznik proved thatB(n, d)is a finite set and asked if, for a fixedk, the set of the polynomials corresponding to each element ofB(n, d)is a constructible subset ofP(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 overQ, i.e. its defining equations are the same for all fieldsk. Moreover, we construct an algorithm that for each pair(n,d )produces a complete list of the elements ofB(n,d )and, for each element of this list, an explicit description of the constructible set of polynomials having this particular Bernstein–Sato polynomial.