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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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