Lipschitz stratifications in o-minimal structures
[Stratifications lipschitziennes dans les structures o-minimales]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 399-421.

Cet article établit l'existence des stratifications lipschitziennes au sens de Mostowski pour les ensembles définissables dans une structure o-minimale polynomialement bornée. On améliore aussi le théorème de préparation de L. van den Dries et P. Speissegger.

This paper establishes existence of Lipschitz stratifications in the sense of Mostowski for sets which are definable in a polynomially bounded o-minimal structure. We also improve L. van den Dries and P. Speissegger's preparation theorem for definable functions.

Publié le :
DOI : 10.24033/asens.2286
Classification : 57N80, 03C64; 32S15, 14P10
Keywords: O-minimal-structures, definable sets, polynomially bounded, Lipschitz geometry, stratifications, regularity conditions, equisingularity.
Mot clés : Structures o-minimales, ensembles définissables, polynomialement borné, géométrie lipschitzienne, stratifications, conditions de régularité, équisingularité. 
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     title = {Lipschitz stratifications  in o-minimal structures},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {399--421},
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     volume = {Ser. 4, 49},
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Nguyen, Nhan; Valette, Guillaume. Lipschitz stratifications  in o-minimal structures. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 399-421. doi : 10.24033/asens.2286. http://archive.numdam.org/articles/10.24033/asens.2286/

Banach, S., Monografie Matematyczne, 17, Polskie Towarzystwo Matematyczne, Warszawa-Wrocław, 1951 | MR

Bochnak, J.; Coste, M.; Roy, M.-F., Ergebn. Math. Grenzg., 12, Springer, Berlin, 1987, 373 pages (ISBN: 3-540-16951-2) | MR | Zbl

Comte, G.; Lion, J.-M.; Rolin, J.-P. Nature log-analytique du volume des sous-analytiques, Illinois J. Math., Volume 44 (2000), pp. 884-888 http://projecteuclid.org/euclid.ijm/1255984698 (ISSN: 0019-2082) | DOI | MR | Zbl

Coste, M. An introduction to o-minimal geometry (2000)

Denef, J.; van den Dries, L. p-adic and real subanalytic sets, Ann. of Math., Volume 128 (1988), pp. 79-138 (ISSN: 0003-486X) | DOI | MR | Zbl

Kaiser, T. On convergence of integrals in o-minimal structures on Archimedean real closed fields, Ann. Polon. Math., Volume 87 (2005), pp. 175-192 (ISSN: 0066-2216) | DOI | MR | Zbl

Kocel-Cynk, B.; Pawłucki, W.; Valette, A. A short geometric proof that Hausdorff limits are definable in any o-minimal structure, Adv. Geom., Volume 14 (2014), pp. 49-58 (ISSN: 1615-715X) | DOI | MR | Zbl

Kurdyka, K.; Parusiński, A., Singularity theory and its applications (Adv. Stud. Pure Math.), Volume 43, Math. Soc. Japan, Tokyo, 2006, pp. 137-177 | DOI | MR | Zbl

Kurdyka, K., Real algebraic geometry (Rennes, 1991) (Lecture Notes in Math.), Volume 1524, Springer, Berlin, 1992, pp. 316-322 | DOI | MR | Zbl

Le Gal, O.; Rolin, J.-P. An o-minimal structure which does not admit C∞ cellular decomposition, Ann. Inst. Fourier (Grenoble), Volume 59 (2009), pp. 543-562 http://aif.cedram.org/item?id=AIF_2009__59_2_543_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Loi, T. L., Singularities and differential equations (Warsaw, 1993) (Banach Center Publ.), Volume 33, Polish Acad. Sci., Warsaw, 1996, pp. 401-409 | MR | Zbl

Loi, T. L. Verdier and strict Thom stratifications in o-minimal structures, Illinois J. Math., Volume 42 (1998), pp. 347-356 http://projecteuclid.org/euclid.ijm/1256045049 (ISSN: 0019-2082) | MR | Zbl

Lion, J.-M.; Rolin, J.-P. Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble), Volume 47 (1997), pp. 859-884 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Mather, J. Notes on topological stability, Bull. Amer. Math. Soc. (N.S.), Volume 49 (2012), pp. 475-506 (ISSN: 0273-0979) | DOI | MR | Zbl

Miller, C. Expansions of the real field with power functions, Ann. Pure Appl. Logic, Volume 68 (1994), pp. 79-94 (ISSN: 0168-0072) | DOI | MR | Zbl

Mostowski, T. Lipschitz equisingularity, Dissertationes Math. (Rozprawy Mat.), Volume 243 (1985), 46 pages (ISSN: 0012-3862) | MR | Zbl

Parusiński, A., Singularities (Warsaw, 1985) (Banach Center Publ.), Volume 20, PWN, Warsaw, 1988, pp. 323-333 | MR | Zbl

Parusiński, A., Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992), Publish or Perish, Houston, TX, 1993, pp. 73-89 | MR | Zbl

Parusiński, A. Lipschitz stratification of subanalytic sets, Ann. Sci. École Norm. Sup., Volume 27 (1994), pp. 661-696 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Pawłucki, W. A linear extension operator for Whitney fields on closed o-minimal sets, Ann. Inst. Fourier (Grenoble), Volume 58 (2008), pp. 383-404 http://aif.cedram.org/item?id=AIF_2008__58_2_383_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Pawłucki, W. Lipschitz cell decomposition in o-minimal structures. I, Illinois J. Math., Volume 52 (2008), pp. 1045-1063 http://projecteuclid.org/euclid.ijm/1254403731 (ISSN: 0019-2082) | DOI | MR | Zbl

Rolin, J.-P.; Speissegger, P.; Wilkie, A. J. Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc., Volume 16 (2003), pp. 751-777 (ISSN: 0894-0347) | DOI | MR | Zbl

Valette, G. Lipschitz triangulations, Illinois J. Math., Volume 49 (2005), pp. 953-979 http://projecteuclid.org/euclid.ijm/1258138230 (ISSN: 0019-2082) | DOI | MR | Zbl

van den Dries, L., London Mathematical Society Lecture Note Series, 248, Cambridge Univ. Press, Cambridge, 1998, 180 pages (ISBN: 0-521-59838-9) | DOI | MR | Zbl

van den Dries, L.; Speissegger, P., Model theory and applications (Quad. Mat.), Volume 11, Aracne, Rome, 2002, pp. 87-116 | MR | Zbl

Łojasiewicz, S.; Stasica, J.; Wachta, K. Stratifications sous-analytiques. Condition de Verdier, Bull. Polish Acad. Sci. Math., Volume 34 (1986), pp. 531-539 (ISSN: 0239-7269) | MR | Zbl

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