Generalised Poincar\'e series and embedded resolution of curves

TL;DR

This paper extends the concepts of generalized Poincaré series for complex curve singularities to curves over perfect fields and relates them to embedded resolutions, broadening their applicability.

Contribution

It introduces a motivic generalization of Poincaré series for curves over perfect fields and expresses these series via embedded resolutions.

Findings

Generalized Poincaré series are extended to perfect fields.

Series are expressed in terms of embedded resolutions.

Broader applicability of motivic invariants for curve singularities.

Abstract

The purpose of this paper is to extend the notions of generalised Poincar\'e series and divisorial generalised Poincar\'e series (of motivic nature) introduced by Campillo, Delgado and Gusein-Zade for complex curve singularities to curves defined over perfect fields, as well as to express them in terms of an embedded resolution of curves.