Poles of the topological zeta function associated to an ideal in dimension two

TL;DR

This paper investigates the poles of the topological zeta function associated with ideals in two variables, providing criteria to identify actual poles and characterizing all possible rational poles.

Contribution

It introduces a criterion to determine whether a candidate pole is genuine and fully describes the set of rational numbers that can occur as poles in this context.

Findings

A criterion for identifying actual poles from candidate poles.

Complete description of possible rational poles for the zeta function.

Results applicable to related zeta functions like the motivic zeta function.

Abstract

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an embedded resolution of the curve. In this paper we will study two questions about the poles of this zeta function. First, we will give a criterion to determine whether or not a candidate pole is a pole. It turns out that we can know this immediately by looking at the intersection diagram of the principalization, together with the numerical data of the exceptional curves. Afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two. The same results are valid for related zeta functions, as for instance the motivic zeta function.