Some remarks on the planar Kouchnirenko's Theorem

TL;DR

This paper investigates various non-degeneracy notions for plane curve singularities, introduces a new weighted homogeneous Newton non-degeneracy, and characterizes when Milnor number and delta-invariant match their Newton diagram formulas, especially in positive characteristic.

Contribution

It establishes the equivalence between classical non-degeneracy conditions and the equality of invariants with their Newton diagram formulas, introducing the new WHNND concept and exploring applications in positive characteristic.

Findings

Equivalence of INND with Milnor number formula

Equivalence of WHNND with delta-invariant formula

Examples illustrating wild vanishing cycles in positive characteristic

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.