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Analytic reparametrization of semi-algebraic sets. (English) Zbl 1143.32007
The paper concerns the controlled parametrizations of compact semi-algebraic sets. The main result is a generalization, to the analytic case, of the following theorem (Yomdin, Gromov). For any natural $$k$$ and any compact semi-algebraic set $$A\subset I^{n}\subset \mathbb{R}^{n}$$ ($$I:=[-1,1]$$) there exists a subdivision of $$A$$ into semi-algebraic pieces $$A_{j},$$ $$j=1,\dots ,N,$$ together with algebraic $$C^{k}$$-mappings $$\psi _{j}:I^{n_{j}}\rightarrow A_{j}$$ such that $$\psi _{j}$$ are onto, homeomorphic on the interior of $$I^{n_{j}}$$ and $$\psi _{j}(x)-\psi _{j}(0)$$ are uniformly bounded by $$1$$ with the all derivatives up to the order $$k.$$ Moreover $$N$$ depends only on $$k$$ and the degrees and the number of equations and inequalities defining $$A.$$ The analytic case differs in that the $$\psi _{j}$$ are real analytic, uniformly bounded with the all derivatives but after removing from $$A$$ a finite number $$M$$ of open boxes of size at most $$2\delta$$ ($$\delta$$ is a given positive real number). Moreover $$M$$ and $$N$$ depends only on $$k$$ and the degrees and the number of equations and inequalities defining $$A$$ times the factor $$\log _{2}(\frac{1}{\delta }).$$

MSC:
 32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions 14P10 Semialgebraic sets and related spaces
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References:
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