## 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

Zbl 0435.32006