Poincar\'e series for plane curve singularities and their behaviour under projections

TL;DR

This paper explores the properties of Poincaré series linked to multi-index filtrations and value semigroups of curve singularities, including their behavior under variable reduction and the role of motivic generalizations.

Contribution

It introduces a comprehensive analysis of Poincaré series for curve singularities, including non-complex cases and motivic generalizations, focusing on variable forgetting properties.

Findings

Poincaré series behavior under variable specialization

Extension to non-complex curve singularities

Inclusion of motivic Poincaré series

Abstract

Our purpose is to investigate all defined Poincar\'e series associated with multi-index filtrations and value semigroups of curve singularities---not necessarily complex---with regard to the property of forgetting variables, i.e., by making variables of the series to be 1. Generalised Poincar\'e series of motivic nature will be also considered.