×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite