The {\L}ojasiewicz Exponent of Semiquasihomogeneous Singularities

TL;DR

This paper provides a formula for the local Łojasiewicz exponent of semiquasihomogeneous functions based on weights, generalizes existing formulas to higher dimensions, and confirms its invariance in certain singularity families, partially affirming Teissier's conjecture.

Contribution

It introduces a new formula for the Łojasiewicz exponent of semiquasihomogeneous functions, extending previous results to n dimensions and establishing invariance in topologically trivial families.

Findings

Derived a formula for the Łojasiewicz exponent in terms of weights.

Extended the formula for quasihomogeneous isolated singularities to n dimensions.

Proved invariance of the Łojasiewicz exponent in certain singularity families.

Abstract

Let $f: (\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ be a semiquasihomogeneous function. We give a formula for the local {\L}ojasiewicz exponent $\mathcal{L}_{0}(f)$ of $f$, in terms of weights of $f$. In particular, in the case of a quasihomogeneous isolated singularity $f$, we generalize a formula for $\mathcal{L}_{0}(f)$ of Krasi\'nski, Oleksik and P{\l}oski ([KOP09]) from $3$ to $n$ dimensions. This was previously announced in [TYZ10], but as a matter of fact it has not been proved correctly there, as noticed by the AMS reviewer T. Krasi\'nski. As a consequence of our result, we get that the {\L}ojasiewicz exponent is invariant in topologically trivial families of singularities coming from a quasihomogeneous germ. This is an affirmative partial answer to Teissier's conjecture. References [KOP09] Tadeusz Krasi\'nski, Grzegorz Oleksik and Arkadiusz P{\l}oski. The {\L}ojasiewicz exponent of an isolated weighted homogeneous surface singularity. Proc. Amer. Math. Soc., 137(10):3387-3397, 2009. [TYZ10] Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo. {\L}ojasiewicz inequality for weighted homogeneous polynomial with isolated singularity. Proc. Amer. Math. Soc., 138(11):3975-3984, 2010.