Milnor number of plane curve singularities in 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.