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Łojasiewicz inequality at singular points. (English) Zbl 1411.32010
Let $$X\subset \mathbb{R}^{n}$$ be a globally subanalytic bounded $$C^{1}$$ submanifold and $$f:X\rightarrow \mathbb{R}$$ a globally subanalytic $$C^{1}$$ function. The set of generalized critical values $$K_{f}$$ of $$f$$ is defined by $K_{f}:=\big\{y\in \mathbb{R}:\exists (x_{k})_{k\in \mathbb{N}}\text{ in }X,\text{ }f(x_{k})\rightarrow y\text{ and }\left| \nabla f(x_{k})\right| \rightarrow 0\big\}.$ The set $$K_{f}$$ is finite and contains the usual critical values of $$f$$ and asymptotic critical values of $$f$$. The main theorem is a variant of the Łojasiewicz inequality: If $$K_{f}\neq \emptyset$$ then there exist a rational number $$\theta \in (0,1)$$ and $$C>0$$ such that for all $$x\in X$$ the following inequality holds $d(f(x),K_{f})^{\theta }\leq C\left| \nabla f(x)\right| .$
##### MSC:
 32B20 Semi-analytic sets, subanalytic sets, and generalizations 58K05 Critical points of functions and mappings on manifolds 14P15 Real-analytic and semi-analytic sets
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##### References:
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