Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity

Tan, Shengli; Yau, Stephen; Zuo, Huaiqing

doi:10.1090/S0002-9939-2010-10387-8

Łojasiewicz inequality for weighted homogeneous polynomial with isolated singularity
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by Shengli Tan, Stephen S.-T. Yau and Huaiqing Zuo PDF

Proc. Amer. Math. Soc. 138 (2010), 3975-3984 Request permission

Abstract:

Let $\nabla f$ be a gradient vector field of a weighted homogenous polynomial with isolated critical point at the origin. Let $(w_1,\dotsc ,w_n)$ be the weights of $f$. In this paper, we prove that the Łojasiewicz Exponent $\theta$ of $f$ is precisely equal to $\displaystyle {\max _{0\leq i\leq n}}w_i-1$. This means that for some constant $c$, $|\nabla f(z)|\geq c|z|^\theta$ in a neighborhood of $0,$ which provides the optimal lower estimate of $|\nabla f(z)|$.

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