Journal of Mathematical Sciences

, Volume 171, Issue 6, pp 736–744 | Cite as

A generalization of the Bombieri–Pila determinant method

  • O. Marmon

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper, we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown’s “Theorem 14” by real-analytic considerations alone. Bibliography: 11 titles.


High Dimension Rational Point Integral Point Projective Variety Plane Curf 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Bohnak, M. Coste,and M.-F. Roy, Géométrie Algébrique Réelle, Springer-Verlag, Berlin (1987).Google Scholar
  2. 2.
    E. Bombieri and J. Pila, “The number of integral points on arcs and ovals,” Duke Math. J., 59, No. 2, 337–357 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    N. Broberg, “A note on a paper by R. Heath-Brown: ‘The density of rational points on curves and surfaces’,” J. reine angew. Math., 571, 159–178 (2004).zbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Burguet, “A proof of Yomdin-Gromov’s algebraic lemma,” Israel J. Math., 168, 291–316 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, New York (1992).zbMATHGoogle Scholar
  6. 6.
    M. Gromov, “Entropy, homology, and semi-algebraic geometry,” Astérisque, 145–146, No. 5, 225–240 (1987).MathSciNetGoogle Scholar
  7. 7.
    D. R. Heath-Brown, “The density of rational points on curves and surfaces,” Ann. Math. (2), 155, No. 2, 553–595 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Pila and A. J. Wilkie, “The rational points of a definable set,” Duke Math. J., 133, No. 3, 591–616 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. Pila, “Integer points on the dilation of a subanalytic surface,” Q. J. Math., 55, No. 2, 207–223 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Salberger, “On the density of rational and integral points on algebraic varieties,” J. reine angew. Math., 606, 123–147 (2007).zbMATHMathSciNetGoogle Scholar
  11. 11.
    Y. Yomdin, “C k-resolution of semi-algebraic mappings. Addendum to: ‘Volume growth and entropy’,” Israel J. Math., 57, No. 3, 301–317 (1987).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Chalmers University of Technology & University of GothenburgGothenburgSweden

Personalised recommendations