# On Maximal order poles of generalized topological zeta functions

## TL;DR

This paper explores examples of topological zeta functions linked to plane curve singularities, revealing cases with multiple second-order poles, contrasting with the standard form where only one such pole exists.

## Contribution

It provides new examples of topological zeta functions with multiple second-order poles, challenging previous understanding of pole multiplicity in these functions.

## Key findings

- Examples of topological zeta functions with several second-order poles.
- Contrasts with standard differential forms where only one second-order pole occurs.
- Highlights the impact of the choice of differential form on pole structure.

## Abstract

We show some examples of topological zeta functions associated to an isolated plane curve singular point and an allowed, in the sense of N\'emethi and Veys, differential form that have several poles of order two. This is in contrast to the case of the standard differential form where, as showed by Veys for plane curves and by Nicaise and Xu in general, there is always at most one pole of order two.