The {\L}ojasiewicz exponent of non-degenerate surface singularities

TL;DR

This paper provides an effective formula for the { extL}ojasiewicz exponent of generic surface singularities using the Newton polyhedron, linking algebraic invariants to topological properties.

Contribution

It introduces a new explicit formula for the { extL}ojasiewicz exponent of generic surface singularities based on their Newton polyhedron.

Findings

Derived an effective formula for the { extL}ojasiewicz exponent

Connected the exponent to the Newton polyhedron of the singularity

Realized one of Arnold's postulates for surface singularities

Abstract

Let $f$ be an isolated singularity at the origin of $\mathbb{C}^n$. One of many invariants that can be associated with $f$ is its {\L}ojasiewicz exponent $\mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for generic surface singularities $f$, an effective formula for $\mathcal{L}_0 (f)$ in terms of the Newton polyhedron of $f$. This is a realization of one of Arnold's postulates.