×

zbMATH — the first resource for mathematics

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

MSC:
14Pxx Real algebraic and real-analytic geometry
58A07 Real-analytic and Nash manifolds
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bochnak, J., Coste, M., Roy, M.-F.: Géométrie-algébrique réelle (Ergebnisse der Mathermatik und ihrer Grenzgebiete, 3. Folge, Band 12). Berlin Heidelberg New York: Springer 1987
[2] Hartshorne, R.: Algebraic geometry. Berlin Heidelberg New York: Springer 1978 · Zbl 0367.14001
[3] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of char=0. Ann. Math.79, 109-326 (1964) · Zbl 0122.38603 · doi:10.2307/1970486
[4] Kurdyka, K., Rusek, K.: Surjectivity of certain injective semialgebraic transformation of ? n Krakòw, preprint (1987) · Zbl 0641.14010
[5] ?ojasiewicz, S.: Ensembles semi-analytiques. Inst. Hautes Etud. Sci. Preprint (1965) · Zbl 0241.32005
[6] Malgrange, B.: Ideals of differentiable functions. Oxford: Oxford University Press 1966 · Zbl 0177.17902
[7] Nash, J.: Real algebraic manifolds. Ann. Math.56, 405-421 (1952) · Zbl 0048.38501 · doi:10.2307/1969649
[8] Paw?ucki, W.: Le théorème de Puiseux pour une application sous-analytique. Bull. Polish. Acad. Sci. Math.32, 555-560 (1984) · Zbl 0574.32010
[9] Risler, J.-J.: Sur l’anneau de fonctions de Nash globales. Ann. Sci. Ec. Norm. Super., IV. Ser8, 365-378 (1975) · Zbl 0318.32002
[10] Walker, R.: Algebraic curves. Princeton: Princeton University Press 1950; Berlin Heidelberg New York: Springer 1978 · Zbl 0039.37701
[11] Zariski, O.: Studies in equisingularity II. Am. J. Math.87, 975-1006 (1965) · Zbl 0146.42502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.