Zeta functions and Bernstein-Sato polynomials for ideals in dimension two

TL;DR

This paper investigates the relationship between topological zeta functions and Bernstein-Sato polynomials for ideals in two dimensions, providing counterexamples and partial positive results to a conjecture about roots and poles.

Contribution

It offers the first counterexamples for monomial and principal ideals in dimension two, challenging a conjecture relating poles of zeta functions to roots of Bernstein-Sato polynomials.

Findings

Counterexamples for monomial ideals in dimension two

Counterexamples for principal ideals in dimension two

Partial positive results on the conjecture

Abstract

For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalization of a conjecture of Igusa - Denef - Loeser predicts that every pole of its topological zeta function is a root of its Bernstein-Sato polynomial. However, typically only a few roots are obtained this way. Following ideas of Veys, we study the following question. Is it possible to find a collection G of polynomials g in C[x_1,...,x_n], such that, for all g in G, every pole of the topological zeta function associated to I and the volume form gdx on the affine n-space, is a root of the Bernstein-Sato polynomial of I, and such that all roots are realized in this way. We obtain a negative answer to this question, providing counterexamples for monomial and principal ideals in dimension two, and give a partial positive result as well.