On the coefficients of the St\"ohr Zeta Function

TL;DR

This paper extends the computation of the Stöhr Zeta Function coefficients for Cohen-Macaulay local rings with multiple branches, providing an upper bound for these coefficients using value semigroups and classification of maximal points.

Contribution

It generalizes previous explicit calculations to rings with any number of branches, offering an upper bound for the coefficients of the zeta function.

Findings

Derived an upper bound for the coefficients of the Stöhr Zeta Function.

Extended explicit computations from one and two branches to multiple branches.

Utilized value semigroup classification to achieve these results.

Abstract

Let $O$ be a one-dimensional Cohen-Macaulay local ring having a finite field as a coefficient field. The aim of this work is to extend the explicit computations of the St\"ohr Zeta Function of $O$ for one and two branches to an arbitrary number of them, obtaining in this general case an upper bound for the coefficients of the zeta function, instead of an equality. The calculations are based on the use of the value semigroup of a curve singularity and a suitable classification of the maximal points of the semigroup.