Duke Mathematical Journal

The rational points of a definable set

J. Pila and A. J. Wilkie

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Abstract

Let XRn be a set that is definable in an o-minimal structure over R. This article shows that in a suitable sense, there are very few rational points of X which do not lie on some connected semialgebraic subset of X of positive dimension

Article information

Source
Duke Math. J., Volume 133, Number 3 (2006), 591-616.

Dates
First available in Project Euclid: 13 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1150201203

Digital Object Identifier
doi:10.1215/S0012-7094-06-13336-7

Mathematical Reviews number (MathSciNet)
MR2228464

Zentralblatt MATH identifier
1217.11066

Subjects
Primary: 11G99: None of the above, but in this section 03C64: Model theory of ordered structures; o-minimality

Citation

Pila, J.; Wilkie, A. J. The rational points of a definable set. Duke Math. J. 133 (2006), no. 3, 591--616. doi:10.1215/S0012-7094-06-13336-7. https://projecteuclid.org/euclid.dmj/1150201203


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References

  • Y. André, ``Arithmetic Gevrey series and transcendence: A survey'' in Les XX IIèmes Journées Arithmetiques (Lille, Belgium, 2001), J. Théor. Nombres Bordeaux 15, Univ. Bordeaux I, Talence, France, 2003, 1--10.
  • G. E. Andrews, An asymptotic expression for the number of solutions of a general class of Diophantine equations, Trans. Amer. Math. Soc. 99 (1961), 272--277.
  • E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 5--42.
  • E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337--357.
  • D. Burguet, A proof of Gromov's algebraic lemma, preprint.
  • L. Van Den Dries, ``Remarks on Tarski's problem concerning $(\bf R, +, \cdot, \exp)$'' in Logic Colloquium '82 (Florence, 1982), Stud. Logic Found. Math. 112, North-Holland, Amsterdam, 1984, 97--121.
  • —, Tame Topology and $O$-Minimal Structures, London Math. Soc. Lecture Note Ser. 248, Cambridge Univ. Press, Cambridge, 1998.
  • L. Van Den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), 183--205.
  • L. Van Den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497--540.
  • G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349--366.
  • A. M. GabrièLov, Projections of semianalytic sets, Funct. Anal. Appl. 2 (1968), 282--291.
  • M. Gromov, Entropy, homology and semialgebraic geometry, Astérisque 145 --.146 (1987), 225--240., Séminaire Bourbaki 1985/86, no. 663.
  • D. R. Heath-Brown, The density of rational points on curves and surfaces, Ann. of Math. (2) 155 (2002), 553--595.
  • M. Hindry and J. H. Silverman, Diophantine Geometry: An Introduction, Grad. Texts in Math. 201, Springer, New York, 2000.
  • V. JarníK, Über die Gitterpunkte auf konvexen Kurven, Math. Z. 24 (1926), 500--518.
  • A. G. Khovanskiǐ, Fewnomials, Transl. Math. Monogr. 88, Amer. Math. Soc., Providence, 1991.
  • S. Lang, Number Theory, III: Diophantine Geometry, Encyclopaedia Math. Sci. 60, Springer, Berlin, 1991.
  • M. Laurent, ``Sur quelques résultats récents de transcendance'' in Journées Arithmétiques 1989 (Luminy, France, 1989), Astérisque 198 --.200, Soc. Math. France, Montrouge, 1991, 209--230.
  • K. Mahler, Lectures on Transcendental Numbers, Lecture Notes in Math. 546, Springer, Berlin, 1976.
  • R. Phillips and P. Sarnak, The spectrum of Fermat curves, Geom. Funct. Anal. 1 (1991), 80--146.
  • J. Pila, Geometric postulation of a smooth function and the number of rational points, Duke Math. J. 63 (1991), 449--463.
  • —, Geometric and arithmetic postulation of the exponential function, J. Austral. Math. Soc. Ser. A 54 (1993), 111--127.
  • —, Integer points on the dilation of a subanalytic surface, Q. J. Math. 55 (2004), 207--223.
  • —, Rational points on a subanalytic surface, Ann. Inst. Fourier (Grenoble) 55 (2005), 1501--1516.
  • —, Note on the rational points of a Pfaff curve, to appear in Proc. Edin. Math. Soc. (2), preprint.
  • A. J. Van Der Poorten, Transcendental entire functions mapping every algebraic number field into itself, J. Austral. Math. Soc. 8 (1968), 192--193.
  • J.-P. Rolin, P. Speissegger, and A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), 751--777.
  • P. C. Sarnak, ``Diophantine problems and linear groups'' in Proceedings of the International Congress of Mathematics, Vol. I (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 459--471.
  • —, Torsion points on varieties and homology of abelian covers, unpublished manuscript.
  • W. M. Schmidt, Integer points on curves and surfaces, Monatsh. Math. 99 (1985), 45--72.
  • —, Integer points on hypersurfaces, Monatsh. Math. 102 (1986), 27--58.
  • H. P. F. Swinnerton-Dyer, The number of lattice points on a convex curve, J. Number Theory 6 (1974), 128--135.
  • M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups, Grundlehren Math. Wiss. 326, Springer, Berlin, 2000.
  • A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), 1051--1094.
  • —, A theorem of the complement and some new o-minimal structures, Selecta Math. (N.S.) 5 (1999), 397--421.
  • —, Diophantine properties of sets definable in o-minimal structures, J. Symbolic Logic 69 (2004), 851--861.
  • Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), 285--300.; addendum: $C^k$-resolution of semialgebraic mappings, Israel J. Math. 57 (1987), 301--317. ;