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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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