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Ensembles semi-algébriques symétriques par arcs. (Semi-algebraic sets symmetric by arcs). (French) Zbl 0686.14027
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 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.
Reviewer: G.Ishikawa

14Pxx Real algebraic and real-analytic geometry
58A07 Real-analytic and Nash manifolds
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
Full Text: DOI EuDML
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