Notre Dame Journal of Formal Logic

Mildness and the Density of Rational Points on Certain Transcendental Curves

G. O. Jones, D. J. Miller, and M. E. M. Thomas

Abstract

We use a result due to Rolin, Speissegger, and Wilkie to show that definable sets in certain o-minimal structures admit definable parameterizations by mild maps. We then use this parameterization to prove a result on the density of rational points on curves defined by restricted Pfaffian functions.

Article information

Source
Notre Dame J. Formal Logic, Volume 52, Number 1 (2011), 67-74.

Dates
First available in Project Euclid: 13 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1292249611

Digital Object Identifier
doi:10.1215/00294527-2010-037

Mathematical Reviews number (MathSciNet)
MR2747163

Zentralblatt MATH identifier
1220.03034

Subjects
Primary: 03C64: Model theory of ordered structures; o-minimality 11G99: None of the above, but in this section 11U09: Model theory [See also 03Cxx]

Keywords
Pfaffian functions parameterization rational points

Citation

Jones, G. O.; Miller, D. J.; Thomas, M. E. M. Mildness and the Density of Rational Points on Certain Transcendental Curves. Notre Dame J. Formal Logic 52 (2011), no. 1, 67--74. doi:10.1215/00294527-2010-037. https://projecteuclid.org/euclid.ndjfl/1292249611


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References

  • [1] Bombieri, E., and W. Gubler, Heights in Diophantine Geometry, vol. 4 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2006.
  • [2] van den Dries, L., Tame Topology and o-Minimal Structures, vol. 248 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1998.
  • [3] van den Dries, L., and P. Speissegger, "The real field with convergent generalized power series", Transactions of the American Mathematical Society, vol. 350 (1998), pp. 4377--421.
  • [4] van den Dries, L., and P. Speissegger, "The field of reals with multisummable series and the exponential function", Proceedings of the London Mathematical Society. Third Series, vol. 81 (2000), pp. 513--65.
  • [5] Gabrielov, A., and N. Vorobjov, "Complexity of computations with Pfaffian and Noetherian functions", pp. 211--50 in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, vol. 137 of NATO Science Series II: Mathematics, Physics and Chemistry, Kluwer Academic Publishers, Dordrecht, 2004.
  • [6] Krantz, S. G., and H. R. Parks, A Primer of Real Analytic Functions, 2d edition, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Boston Inc., Boston, 2002.
  • [7] Miller, C., "Infinite differentiability in polynomially bounded o-minimal structures", Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2551--55.
  • [8] Pila, J., "Mild parameterization and the rational points of a Pfaff curve", Commentarii Mathematici Universitatis Sancti Pauli, vol. 55 (2006), pp. 1--8.
  • [9] Pila, J., "The density of rational points on a Pfaff curve", Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6, vol. 16 (2007), pp. 635--45.
  • [10] Pila, J., "Counting rational points on a certain exponential-algebraic surface", Université de Grenoble. Annales de l'Institut Fourier, vol. 60 (2010), pp. 489--514.
  • [11] Rolin, J.-P., P. Speissegger, and A. J. Wilkie, "Quasianalytic Denjoy-Carleman classes and o-minimality", Journal of the American Mathematical Society, vol. 16 (2003), pp. 751--77.
  • [12] Wilkie, A. J., "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function", Journal of the American Mathematical Society, vol. 9 (1996), pp. 1051--94.