Ensembles semi-algébriques symétriques par arcs. (Semi-algebraic sets symmetric by arcs)
The author defines and studies a natural and interesting class of semi- algebraic sets, that is, the class of semi-algebraic sets symmetric by arcs. A semi-algebraic set E in \({\mathbb{R}}^ n\) is called \textit{symmetric by arcs} if, for any analytic arc \(\gamma:\quad (-1,1)\to {\mathbb{R}}^ n,\) \(\gamma\) ((-1,0))\(\subset E\) implies \(\gamma\) ((-1,1))\(\subset E\). Then, define AR-topology of \({\mathbb{R}}^ n\) such that \(E\subset {\mathbb{R}}^ n\) is closed if and only if E is a semi-algebraic set symmetric by arcs. In this paper, for example, it is given a characterization of connected components of desingularization of an algebraic set, in the ward of AR- irreducible decomposition. - Further, the author defines the notion of arc-analytic mapping and he studies, for a semi-algebraic set X symmetric by arcs, the structure of the ring \(A_ a(X)\) of functions \(X\to {\mathbb{R}}\) semi-algebraic and arc-analytic.
scientific article; zbMATH DE number 4123883