Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-15T14:47:08.025Z Has data issue: false hasContentIssue false
  • English
  • Français

$\mathscr{C}^{p}$-parametrization in O-minimal Structures

Part of:Model theory

Published online by Cambridge University Press:  09 January 2019

Beata Kocel-Cynk,
Wiesław Pawłucki and
Anna Valette
Beata Kocel-Cynk
Affiliation:
Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, PL31-155 Cracow, Poland Email: bkocel@usk.pk.edu.pl
Wiesław Pawłucki
Affiliation:
Institute of Mathematics, Jagiellonian University, ul. St. Łojasiewicza 6, PL30-348 Cracow, Poland Email: wieslaw.pawlucki@im.uj.edu.planna.valette@im.uj.edu.pl
Anna Valette
Affiliation:
Institute of Mathematics, Jagiellonian University, ul. St. Łojasiewicza 6, PL30-348 Cracow, Poland Email: wieslaw.pawlucki@im.uj.edu.planna.valette@im.uj.edu.pl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We give a geometric and elementary proof of the uniform $\mathscr{C}^{p}$-parametrization theorem of Yomdin and Gromov in arbitrary o-minimal structures.

Keywords

o-minimal structureCp-parametrization

MSC classification

Primary:03C64: Model theory of ordered structures; o-minimality
Secondary:14P15: Real analytic and semianalytic sets32B20: Semi-analytic sets and subanalytic sets

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019, pp. 99 - 108
Copyright
© Canadian Mathematical Society 2018 

References

Burguet, D., A proof of Yomdin-Gromov’s algebraic lemma . Israel J. Math. 168(2008), 291–316. https://doi.org/10.1007/s11856-008-1069-z.Google Scholar
Cluckers, R., Comte, G., and Loeser, F., Non-archimedean Yomdin-Gromov parametrization and points of bounded height. 2014. arxiv:1404.1952.Google Scholar
Cluckers, R., Pila, J., and Wilkie, A., Uniform parametrization of subanalytic sets and diophantine applications. 2018. arxiv:1605.05916.Google Scholar
Coste, M., An introduction to O-minimal geometry, Dottorato di Ricerca in Matematica, Edizioni ETS, Pisa, 2000.Google Scholar
Gromov, M., Entropy, homology and semialgebraic geometry . Séminaire Bourbaki, 1985/86, Astérisque 145–146(1987), 5, 225–240.Google Scholar
Hironaka, H., Introduction to real analytic sets and real analytic maps. Dottorato di Ricerca in Matematica, Edizioni ETS, Pisa, 2009.Google Scholar
Kocel-Cynk, B., Pawłucki, W., and Valette, A., Short geometric proof that Hausdorff limits are definable in any o-minimal structure . Adv. Geom. 14(2014), no. 1, 49–58. https://doi.org/10.1515/advgeom-2013-0028.Google Scholar
Kurdyka, K. and Pawłucki, W., Subanalytic version of Whitney’s extension theorem . Studia Math. 124(1997), no. 3, 269–280.Google Scholar
Łojasiewicz, S., Ensembles semi-analytiques. Institut des Hautes Études Scientifiques, Bures-sur-Yvette, 1965.Google Scholar
Pawłucki, W., Le théorème de Puiseux pour une application sous-analytique . Bull. Polish Acad. Sci. Math. 32(1984), no. 9–10, 555–560.Google Scholar
Pawłucki, W., Lipschitz cell decomposition in o-minimal structures. I . Illinois J. Math. 52(2008), 1045–1063.Google Scholar
Pila, J. and Wilkie, A. J., The rational points of a definable set . Duke Math. J. 133(2006), no. 3, 591–616. https://doi.org/10.1215/S0012-7094-06-13336-7.Google Scholar
Valette, G., Lipschitz triangulations . Illinois J. Math. 49(2005), no. 3, 953–979.Google Scholar
van den Dries, L., Tame topology and o-minimal structures. London Mathematical Society Lecture Note Series, 248, Cambridge University Press, Cambridge, 1998. https://doi.org/10.1017/CBO9780511525919.Google Scholar
Yomdin, Y., Volume growth and entropy . Israel J. Math. 57(1987), 285–300. https://doi.org/10.1007/BF02766215.Google Scholar
Yomdin, Y., C k -resolution of semialgebraic mappings. Addendum to: “Volume growth and entropy” . Israel J. Math. 57(1987), 301–317. https://doi.org/10.1007/BF02766216.Google Scholar
Yomdin, Y., Analytic reparametrization of semialgebraic sets . J. Complexity 24(2008), no. 1, 54–76. https://doi.org/10.1016/j.jco.2007.03.009.Google Scholar
Yomdin, Y., Smooth parametrizations in dynamics, analysis, diophantine and computational geometry . Jpn. J. Ind. Appl. Math. 32(2015), no. 2, 411–435. https://doi.org/10.1007/s13160-015-0176-6.Google Scholar