# On the Milnor formula in arbitrary characteristic

## TL;DR

This paper explores the validity of the Milnor formula in arbitrary characteristic, identifying conditions under which it holds and improving existing results on the Milnor number of irreducible singularities.

## Contribution

It extends the understanding of the Milnor formula in characteristic p, providing new conditions for its validity and refining previous bounds on the Milnor number.

## Key findings

- Milnor formula holds for Newton non-degenerate singularities in characteristic p.
- Milnor formula holds if p exceeds the intersection number with the generic polar.
- Improved bounds on the Milnor number for irreducible singularities.

## Abstract

The Milnor formula $\mu=2\delta-r+1$ relates the Milnor number $\mu$, the double point number $\delta$ and the number $r$ of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality $\mu\geq 2\delta-r+1$ in arbitrary characteristic and showed that the equality $\mu=2\delta-r+1$ characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic $p$. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. (2010) 25) or if $p$ is greater than the intersection number of the singularity with its generic polar (Nguyen H.D., Annales de l'Institut Fourier, Tome 66 (5) (2016)). Then we improve our result on the Milnor number of irreducible singularities (Bull. London Math. Soc. 48 (2016)). Our considerations are based on the properties of polars of plane singularities in characteristic $p$.