Journal of the American Mathematical Society

Author(s): J.-P. Rolin; P. Speissegger; A. J. Wilkie
Journal: J. Amer. Math. Soc. 16 (2003), 751-777.
MSC (2000): Primary 14P15, 03C64; Secondary 32S45
Posted: March 21, 2003
Retrieve article in: PDF

Abstract: We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure conditions. Some of these structures do not admit analytic cell decomposition, and they show that there is no largest o-minimal expansion of the real field.

1.: S. S. Abhyankar and T. T. Moh, Newton-Puiseux expansion and generalized Tschirnhausen transformation I, J. Reine Angew. Math., 260 (1973), pp. 47-83. MR 49:2724
2.: -, Newton-Puiseux expansion and generalized Tschirnhausen transformation II, J. Reine Angew. Math., 261 (1973), pp. 29-54. MR 49:2724
3.: E. Bierstone and P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), pp. 5-42. MR 89k:32011
4.: -, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., 128 (1997), pp. 207-302. MR 98e:14010
5.: C. L. Childress, Weierstrass division in quasianalytic local rings, Canad. J. Math., 28 (1976), pp. 938-953. MR 54:5491
6.: L. van den Dries, O-minimal structures and real analytic geometry, in Current developments in mathematics, 1998, International Press, 1999, pp. 105-152. MR 2001j:03075
7.: L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J., 84 (1996), pp. 497-540. MR 97i:32008
8.: L. van den Dries and P. Speissegger, The real field with convergent generalized power series is model complete and o-minimal, Trans. Amer. Math. Soc., 350 (1998), pp. 4377-4421. MR 99a:03036
9.: -, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3), 81 (2000), pp. 513-565. MR 2002k:03057
10.: A. Gabrielov, Complements of subanalytic sets and existential formulas for analytic functions, Invent. Math., 125 (1996), pp. 1-12. MR 97h:32007
11.: Y. Katznelson, Introduction to Harmonic Analysis, John Wiley, 1968. MR 40:1734
12.: H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Japan Acad., Ser. A, 55 (1979), pp. 69-72. MR 80e:58007
13.: J.-M. Lion and J.-P. Rolin, Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier, 47 (1997), pp. 859-884. MR 98h:32009
14.: S. Mandelbrojt, Sur les fonctions indéfiniment dérivables, Acta Math., 72 (1940), pp. 15-29. MR 1:297d
15.: C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic, 68 (1994), pp. 79-94. MR 95i:03081
16.: -, Infinite differentiability in polynomially bounded o-minimal structures, Proc. Amer. Math. Soc., 123 (1995), pp. 2551-2555. MR 95j:03069
17.: A. Parusinski, Lipschitz stratification of subanalytic sets, Ann. Sci. Ecole Norm. Sup. (4), 27 (1994), pp. 661-996. MR 96g:32017
18.: C. Roumieu, Ultradistributions définies sur $\mathbb{R} n$ et sur certaines classes de variétés différentiables, J. Anal. Math., 10 (1962-1963), pp. 153-192. MR 28:1487
19.: W. Rudin, Real and Complex Analysis, McGraw-Hill, 1987. MR 88k:00002

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14P15, 03C64, 32S45

J.-P. Rolin
Affiliation: Laboratoire de Topologie, Université de Bourgogne, 9 Av. Alain Savary, B.P. 47870, 21078 Dijon Cedex, France
Email: rolin@u-bourgogne.fr

P. Speissegger
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: speisseg@math.wisc.edu

A. J. Wilkie
Affiliation: Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford OX1 3LB, United Kingdom
Email: wilkie@maths.ox.ac.uk

DOI: 10.1090/S0894-0347-03-00427-2
PII: S 0894-0347(03)00427-2
Keywords: Quasianalytic classes, o-minimal structures, resolution of singularities
Received by editor(s): February 19, 2001
Posted: March 21, 2003
Additional Notes: Supported in part by CNRS, NSERC grant OGP0009070 and NSF grant DMS-9988453
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1088-6834(e) ISSN: 0894-0347(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS