no. 5
Nash triviality in families of Nash mappings
[Trivialité de Nash dans des familles d'applications de Nash]
Escribano, Jesús1
1 Universidad Complutense, Facultad de CC Matematcias, Departamento de Sistemas Informáticos y Programací on, 28040 Madrid (Espagne)
Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1209-1228.

On étudie des résultats de trivialité pour les familles de Nash de submersions de Nash propres. On travaille avec des variétés et des applications de Nash définies sur un corps réel clos quelconque R. Pour éviter l’usage de l’intégration des champs de vecteurs (processus non algébrique en général), on étudie les fibres de telles familles sur les points appartenant au spectre réel R p ˜, et on construit des modèles de submersions de Nash propres sur des corps réels clos plus petits. Finalement, on obtient des résultats de finitude sur le type topologique des familles d’applications de Nash, et aussi des résultats sur l’effectivité de telles constructions.

We study triviality of Nash families of proper Nash submersions or, in a more general setting, the triviality of pairs of proper Nash submersions. We work with Nash manifolds and mappings defined over an arbitrary real closed field R. To substitute the integration of vector fields, we study the fibers of such families on points of the real spectrum R p ˜ and we construct models of proper Nash submersions over smaller real closed fields. Finally we obtain results on finiteness of topological types in families of Nash mappings, and also results on effectiveness of the above constructions.

DOI : 10.5802/aif.1852
Classification : 14P20, 58A07
Keywords: Nash manifold, Nash mapping, Nash triviality, real spectrum
Mot clés : variété de Nash, application de Nash, trivialité de Nash, spectre réel
Escribano, Jesús 1

1 Universidad Complutense, Facultad de CC Matematcias, Departamento de Sistemas Informáticos y Programací on, 28040 Madrid (Espagne) 
@article{AIF_2001__51_5_1209_0,
     author = {Escribano, Jes\'us},
     title = {Nash triviality in families of {Nash} mappings},
     journal = {Annales de l'Institut Fourier},
     pages = {1209--1228},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {5},
     year = {2001},
     doi = {10.5802/aif.1852},
     mrnumber = {1860663},
     zbl = {0979.14026},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1852/}
}
TY  - JOUR
AU  - Escribano, Jesús
TI  - Nash triviality in families of Nash mappings
JO  - Annales de l'Institut Fourier
PY  - 2001
SP  - 1209
EP  - 1228
VL  - 51
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1852/
DO  - 10.5802/aif.1852
LA  - en
ID  - AIF_2001__51_5_1209_0
ER  - 
%0 Journal Article
%A Escribano, Jesús
%T Nash triviality in families of Nash mappings
%J Annales de l'Institut Fourier
%D 2001
%P 1209-1228
%V 51
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1852/
%R 10.5802/aif.1852
%G en
%F AIF_2001__51_5_1209_0
Escribano, Jesús. Nash triviality in families of Nash mappings. Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1209-1228. doi : 10.5802/aif.1852. http://www.numdam.org/articles/10.5802/aif.1852/

[BCR] J. Bochnak; M. Coste; M-F. Roy Real Algebraic Geometry, Ergeb. Math. Grenzgeb., (3) 36, Springer-Verlag, Berlin - Heidelberg - New York, 1998 | MR | Zbl

[CS1] M. Coste; M. Shiota Nash triviality in families of Nash manifolds, Invent. Math, Volume 108 (1992), pp. 349-368 | DOI | MR | Zbl

[CS2] M. Coste; M. Shiota; Ed. F. Broglia, M. Galbiati Thom's first isotopy lemma: a semialgebraic version with uniform bound, Real Analytic and Algebraic Geometry (1995), pp. 83-101 | Zbl

[DK] H. Delfs; M. Knebush Locally Semialgebraic Spaces, Lecture Notes in Mathematics, 1173, Springer-Verlag, Berlin, 1985 | MR | Zbl

[GWPL] C.G. Gibson; K. Wirthmüller; A.A. du Plessis; E.J.N. Looijenga Topological stability of smooth mappings, Lecture Notes in Mathematics, 552, Springer-Verlag, Berlin, 1976 | MR | Zbl

[H] R. Hardt Semi-algebraic local triviality in semi-algebraic mappings, Amer. J. Math., Volume 102 (1980), pp. 291-302 | DOI | MR | Zbl

[R] R. Ramanakoraisina Complexité des fonctions de Nash, Commun. Algebra, Volume 17 (1989) no. 6, pp. 1395-1406 | DOI | MR | Zbl

[Shi1] M. Shiota Nash Manifolds, Lecture Notes in Mathematics, 1269, Springer-Verlag, Berlin, 1987 | MR | Zbl

[Shi2] M. Shiota Geometry of Subanalytic and Semialgebraic Sets, Progress in Mathematics, vol. 150, Birkhäuser, Boston, 1997 | MR | Zbl

Cité par Sources :