Bernstein-Sato polynomials for general ideals vs. principal ideals

Abstract

We show that given an ideal a \mathfrak {a} generated by regular functions f 1 , … , f r f_1,\ldots ,f_r on X X , the Bernstein-Sato polynomial of a \mathfrak {a} is equal to the reduced Bernstein-Sato polynomial of the function g = ∑ i = 1 r f i y i g=\sum _{i=1}^rf_iy_i on X × A r X\times \mathbf {A}^r . By combining this with results from Budur, Mustaţă, and Saito [Compos. Math. 142 (2006), pp. 779–797], we relate invariants and properties of a \mathfrak {a} to those of g g . We also use the result on Bernstein-Sato polynomials to show that the Strong Monodromy Conjecture for Igusa zeta functions of principal ideals implies a similar statement for arbitrary ideals.