Milnor number of plane curve singularities in arbitrary characteristic

TL;DR

This paper generalizes Kouchnirenko's formula and a conjecture related to the Milnor number of plane curve singularities over fields of arbitrary characteristic, extending known results beyond characteristic zero.

Contribution

It extends classical formulas and conjectures to arbitrary characteristic, providing new proofs and partial results for the Milnor number of plane curve singularities.

Findings

Generalized Kouchnirenko's formula for degenerated power series

Proved the conjecture in some cases using Greuel and Nguyen's methods

Extended the relation between normalization and jacobian ideal to arbitrary characteristic

Abstract

Reduced power series in two variables with coefficients in a field of characteristic zero satisfy a well-known formula that relates a codimension related to the normalization of a ring and the jacobian ideal. In the general case Deligne proved that this formula is only an inequality; Garc\'ia Barroso and P{\l}oski stated a conjecture for irreducible power series. In this work we generalize Kouchnirenko's formula for any degenerated power series and also generalize Garc\'ia Barroso and P{\l}oski's conjecture. We prove the conjecture in some cases using in particular Greuel and Nguyen.