The Milnor number of plane irreducible singularities in positive characteristic

TL;DR

This paper characterizes when the Milnor number equals the degree of the conductor for irreducible plane curve singularities over fields of positive characteristic, using semigroup conditions.

Contribution

It provides necessary and sufficient conditions for the equality of Milnor number and conductor degree in positive characteristic, extending classical results to this setting.

Findings

Conditions for $=$ in terms of semigroup

Results valid when characteristic $p$ exceeds the order of $f$

Clarifies the relationship between Milnor number and conductor in positive characteristic

Abstract

\noindent Let $\mu(f)$ resp. $c(f)$ be the Milnor number resp. the degree of the conductor of an irreducible power series $f\in \bK[[x,y]]$, where $\bK$ is an algebraically closed field of characteristic $p\geq 0$. It is well-known that $\mu(f)\geq c(f)$. We give necessary and sufficient conditions for the equality $\mu(f)=c(f)$ in terms of the semigroup associated with $f$, provided that $p>\ord f$.