Some remarks on the planar Kouchnirenko's Theorem

Abstract

We consider different notions of non-degeneracy, as introduced by Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve singularities $\{f(x,y) = 0\}$ and introduce the new notion of weighted homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number $\mu$ resp. the delta-invariant $\delta$ can be computed by explicit formulas $\mu_N$ resp. $\delta_N$ from the Newton diagram of $f$ if $f$ is NND resp. WNND. It was however unknown whether the equalities $\mu=\mu_N$ resp. $\delta=\delta_N$ can be characterized by a certain non-degeneracy condition on $f$ and, if so, by which one. We show that $\mu=\mu_N$ resp. $\delta=\delta_N$ is equivalent to INND resp. WHNND and give some applications and interesting examples related to the existence of "wild vanishing cycles". Although the results are new in any characteristic, the main difficulties arise in positive characteristic.