A note on the plane curve singularities in positive characteristic

Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\ge 0$ we consider the Milnor number $\mu (f)$, the delta invariant $\delta (f)$ and the number $r(f)$ of its irreducible components. Put $\bar{\mu }(f)=2\delta (f)-r(f)+1$. If $p=0$ then $\bar{\mu }(f)=\mu (f)$ (the Milnor formula). If $p>0$ then $\mu (f)$ is not an invariant and $\bar{\mu }(f)$ plays the role of $\mu (f)$. Let ${\mathcal{N}}_f$ be the Newton polygon of $f$. We define the numbers $\mu ({\mathcal{N}}_f)$ and $r({\mathcal{N}}_f)$ which can be computed by explicit formulas. The aim of this note is to give a simple proof of the inequality $\bar{\mu }(f)-\mu ({\mathcal{N}}_f)\ge r({\mathcal{N}}_f)-r(f)\ge 0$ due to Boubakri, Greuel and Markwig. We also prove that $\bar{\mu }(f)=\mu ({\mathcal{N}}_f)$ when $f$ is non-degenerate.