On bernstein-sato polynomials

We show that for fixed n and d the set of Bernstein-Sato polynomials of all the polynomials in at most n variables of degrees at most d is finite. As a corollary, we show that there exists an integer t depending only on n and d such that f^-t generates R_f as a module over the ring of the k-linear differential operators of R, where k is an arbitrary field of characteristic 0, R is the ring of polynomials in n variables over k and f ∈ R is an arbitrary non-zero polynomial of degree at most d.