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Bound of the metric invariant for the Noetherian family. (Borne d’invariant métrique pour une famille noethérienne.) (French) Zbl 0933.32012
Summary: The motivation of this paper is a question asked by B. Teissier in Lect. Notes Math. 431, 295-317 (1975; Zbl 0333.32009): if $$\varphi:M\to N$$ is an analytic morphism between two real analytic manifolds $$M$$ and $$N$$, and if $$K$$ is a compact subanalytic set of $$M$$, then for every point $$x_0$$ in $$\varphi(K)$$ there exists an open neighbourhood $$U$$ of $$x_0$$ in $$N$$ and a constant $$\gamma>0$$ such that for all $$x$$ in $$U$$ and all $$(a,b)$$ in the same connected component of $$\varphi^{-1}(x)\cap K$$, there exist a rectifiable curve in $$\varphi^{-1} (x)\cap K$$ joining $$a$$ and $$b$$ with length less than $$\gamma$$.
In this paper we prove the following statement: let $$\Omega$$ be an open set of $$\mathbb{R}^n$$, $$N$$ a real analytic manifold, $$\varphi: \Omega\to N$$ a proper analytic morphism and $$K\subset\Omega$$ an analytic subset of $$\Omega$$. Then for every point $$y_0$$ of $$N$$ there are an open neighbourhood $$U$$ in $$N$$ and a constant $$\eta>0$$ such that for all $$y$$ in $$U$$ there exists $$C_y>0$$ satisfying the following: for every points $$a,b$$ of the same connected component of $$\varphi^{-1} (y)\cap K$$ there exists an analytic rectifiable curve $$\sigma$$ in $$\varphi^{-1} (y)\cap K$$ joining $$a$$ and $$b$$ with $$|\sigma |\leq C_y|a-b|^\eta$$, where $$|\sigma|$$ is the length of $$\sigma$$ and $$|a-b|$$ is the euclidean distance between $$a$$ and $$b$$.
##### MSC:
 32B05 Analytic algebras and generalizations, preparation theorems 32B10 Germs of analytic sets, local parametrization 32B20 Semi-analytic sets, subanalytic sets, and generalizations
##### Keywords:
real analytic manifold; proper analytic morphism