On a Newton filtration for functions on a curve singularity

TL;DR

This paper introduces a new method to compute the Poincaré series for plane curve singularities, demonstrating that it depends solely on the Newton diagram, regardless of the specific defining equation.

Contribution

It presents an alternative technique for calculating the Poincaré series without the non-degeneracy assumption, extending previous results to all plane curve singularities.

Findings

Poincaré series depends only on the Newton diagram

New technique applicable to all plane curve singularities

Series calculation independent of the defining equation

Abstract

In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincar\'e series was computed for plane curve singularities non-degenerate with respect to their Newton diagrams. Here we use another technique to compute the Poincar\'e series for plane curve singularities without the assumption that they are non-degenerate with respect to their Newton diagrams. We show that the Poincar\'e series only depends on the Newton diagram and not on the defining equation.