The {\L}ojasiewicz Exponent of Semiquasihomogeneous Singularities

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.