On the {\L}ojasiewicz exponent, special direction and maximal polar quotient

TL;DR

This paper characterizes special directions in plane curve singularities where the Lojasiewicz exponent isn't attained on the polar curve, and relates this to the maximal polar quotient being less than its generic value, using Eggers trees.

Contribution

It provides a complete characterization of nonsingular directions affecting the Lojasiewicz exponent and introduces the use of Eggers trees to analyze singularity relations.

Findings

Identifies all nonsingular directions where the Lojasiewicz exponent is not attained on the polar curve.

Shows that for non-Morse singularities, the maximal polar quotient is strictly less than the generic value for these directions.

Utilizes Eggers trees to analyze the structure of singularities and their polar relations.

Abstract

For a local singular plane curve germ $f(X,Y)=0$ we characterize all nonsingular $\lambda\in\bbC\{X,Y\}$ such that the {\L}ojasiewicz exponent of $\grad\,f$ is not attained on the polar curve $\bJ(\lambda,f)=0$. When $f$ is not Morse we prove that for the same $\lambda$'s the maximal polar quotient $q_0(f,\lambda)$ is strictly less than its generic value $q_0(f)$. Our main tool is the Eggers tree of singularity constructed as a decorated graph of relations between balls in the space of branches defined by using a logarithmic distance.