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January, 2021 Effective Łojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs
Beata OSIŃSKA-ULRYCH, Grzegorz SKALSKI, Stanisław SPODZIEJA
J. Math. Soc. Japan 73(1): 277-299 (January, 2021). DOI: 10.2969/jmsj/83378337

Abstract

Let $X \subset \mathbf{\mathbb{R}}^n$ be a compact semialgebraic set and let $f : X \to \mathbf{\mathbb{R}}$ be a nonzero Nash function. We give a Solernó and D'Acunto–Kurdyka type estimation of the exponent $\varrho\in[0,1)$ in the Łojasiewicz gradient inequality $|\nabla f(x)| \ge C|f(x)|^\varrho$ for $x \in X$, $|f(x)| < \varepsilon$ for some constants $C,\varepsilon > 0$, in terms of the degree of a polynomial $P$ such that $P(x,f(x)) = 0$, $x \in X$. As a corollary we obtain an estimation of the degree of sufficiency of non-isolated Nash function singularities.

Citation

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Beata OSIŃSKA-ULRYCH. Grzegorz SKALSKI. Stanisław SPODZIEJA. "Effective Łojasiewicz gradient inequality for Nash functions with application to finite determinacy of germs." J. Math. Soc. Japan 73 (1) 277 - 299, January, 2021. https://doi.org/10.2969/jmsj/83378337

Information

Received: 25 September 2019; Published: January, 2021
First available in Project Euclid: 1 December 2020

Digital Object Identifier: 10.2969/jmsj/83378337

Subjects:
Primary: 14R99
Secondary: 14P05, 32S70

Rights: Copyright © 2021 Mathematical Society of Japan

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Vol.73 • No. 1 • January, 2021
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