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Le théorème de Puiseux pour une application sous-analytique. (French) Zbl 0574.32010
The following result is proved: ”Let M be a bounded m-dimensional real analytic (locally closed) submanifold of \({\mathbb{R}}^ n\) which is also subanalytic. Suppose that \(f: M\times (0,1)\to {\mathbb{R}}^ p\) is a bounded real analytic map with subanalytic graph in \({\mathbb{R}}^ n\times {\mathbb{R}}\times {\mathbb{R}}^ p\). Then there exists a set E and an integer k such that: (i) E is closed in M and subanalytic in \({\mathbb{R}}^ n\); (ii) dim \(E\leq m-1\); (iii) for each \(a\in M\setminus E\) the map \((x,t)\to f(x,t^ k)\) has an analytic extension to a neighbourhood of (a,0) in \(M\times {\mathbb{R}}.''\)
The proof uses the techniques developed by Lojasiewicz and his collaborators [see Z. Denkowska, S. Łojasiewicz and J. Stasica, Bull. Acad. Pol. Sci., Sér. Sci. Math. 27, 529-536 (1979; Zbl 0435.32006), and S. Lojasiewicz, ”Ensembles semi-analytiques” (Inst. Haut. Étud. Sci., Bures-sur-Yvette 1965)].
Reviewer: N.Mihalache

MSC:
32B20 Semi-analytic sets, subanalytic sets, and generalizations
32D15 Continuation of analytic objects in several complex variables
32C05 Real-analytic manifolds, real-analytic spaces
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