The Łojasiewicz exponent of a continuous subanalytic function at an isolated zero

Sơn, Phạm

doi:10.1090/S0002-9939-2010-10683-4

The Łojasiewicz exponent of a continuous subanalytic function at an isolated zero
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Proc. Amer. Math. Soc. 139 (2011), 1-9 Request permission

Corrigendum: Proc. Amer. Math. Soc. 148 (2020), 2739-2741.

Abstract:

Let $f$ be a continuous subanalytic function defined in a neighborhood of the origin $0 \in \mathbb R^n$ such that $f$ has an isolated zero at $0.$ We describe the smallest possible exponents $\alpha , \beta , \theta$ for which we have the following estimates: $|f(x)| \ge c \|x\|^\alpha , {\frak m}_f(x) \ge c \|x\|^\beta , {\frak m}_f(x) \ge c |f(x)|^\theta$ for $x$ near zero, where $c > 0$ and ${\frak m}_f(x)$ is the nonsmooth slope of $f$ at $x.$ We prove that $\alpha = \beta + 1, \theta = \beta /\alpha .$ In the smooth case, we have ${\frak m}_f(x) = \|\nabla f(x) \|,$ and we therefore retrieve a result of Gwoździewicz, which is a counterpart of the result of Teissier in the complex case.

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