Counting rational points on a certain exponential-algebraic surface
Annales de l'Institut Fourier, Volume 60 (2010) no. 2, p. 489-514
We study the distribution of rational points on a certain exponential-algebraic surface and we prove, for this surface, a conjecture of A. J. Wilkie.
Nous étudions la répartition des points rationnels sur une certaine surface exponentielle-algébrique et prouvons, pour cette surface, une conjecture de A. J. Wilkie.
DOI : https://doi.org/10.5802/aif.2530
Classification:  11G99,  03C64
Keywords: O-minimal structure, rational points, transcendental numbers
@article{AIF_2010__60_2_489_0,
     author = {Pila, Jonathan},
     title = {Counting rational points on a certain exponential-algebraic surface},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {2},
     year = {2010},
     pages = {489-514},
     doi = {10.5802/aif.2530},
     zbl = {1210.11074},
     mrnumber = {2667784},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2010__60_2_489_0}
}
Counting rational points on a certain exponential-algebraic surface. Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 489-514. doi : 10.5802/aif.2530. https://aif.centre-mersenne.org/item/AIF_2010__60_2_489_0/

[1] Bombieri, E. Algebraic values of meromorphic maps, Inventiones, Tome 10 (1970), pp. 267-287 | Article | MR 306201 | Zbl 0214.33702

[2] Bombieri, E.; Gubler, W. Heights in Diophantine geometry, New Mathematical Monographs 4. Cambridge: Cambridge University Press. xvi, 652 p. (2007) | MR 2216774 | Zbl 1115.11034

[3] Bombieri, E.; Pila, J. The number of integral points on arcs and ovals, Duke Math. J., Tome 59 (1989), pp. 337-357 | Article | MR 1016893 | Zbl 0718.11048

[4] Butler, L. Some cases of Wilkie’s conjecture (2009) (working paper)

[5] Van Den Dries, Lou Tame topology and o-minimal structures, London Mathematical Society, Lecture Note Series. 248. Cambridge University Press, Cambridge: x, 180 p (1998) | MR 1633348 | Zbl 0953.03045

[6] Van Den Dries, Lou; Miller, C. Geometric categories and o-minimal structures, Duke Math. J., Tome 84 (1996) no. 2, pp. 497-540 | Article | MR 1404337 | Zbl 0889.03025

[7] Gabrielov, A.; Vorobjov, N. Complexity of computations with Pfaffian and Noetherian functions, Normal Forms, Bifurcations and Finiteness problems in Differential Equations, Kluwer (2004) | MR 2083248

[8] Lang, S. Algebraic values of meromorphic functions, Topology, Tome 3 (1965), pp. 183-191 | Article | MR 190092 | Zbl 0133.13804

[9] Lang, S. Introduction to transcendental numbers, Addison-Wesley, Reading Mass (1966) | MR 214547 | Zbl 0144.04101

[10] Pila, J. Integer points on the dilation of a subanalytic surface, Q. J. Math., Tome 55 (2004) no. 2, pp. 207-223 | Article | MR 2068319 | Zbl 1111.32004

[11] Pila, J. Rational points on a subanalytic surface, Ann. Inst. Fourier, Tome 55 (2005) no. 5, pp. 1501-1516 | Article | Numdam | MR 2172272 | Zbl 1121.11032

[12] Pila, J. Mild parameterization and the rational points of a pfaff curve, Commentarii Mathematici Universitatis Sancti Pauli, Tome 55 (2006), pp. 1-8 (and Erratum p.231) | MR 2251995 | Zbl 1129.11029

[13] Pila, J. The density of rational points on a pfaff curve, Ann. Fac. Sci. Toulouse, Tome 16 (2007), pp. 635-645 | Article | Numdam | MR 2379055

[14] Pila, J. On the algebraic points of a definable set, Selecta Math. N.S., Tome 15 (2009), pp. 151-170 | Article | MR 2511202

[15] Pila, J.; Wilkie, A. J. The rational points of a definable set, Duke Math. J., Tome 133 (2006), pp. 591-616 | Article | MR 2228464 | Zbl pre05043321

[16] Roy, D. Interpolation formulas and auxiliary functions, J. Number Theory, Tome 94 (2002), pp. 248-285 | Article | MR 1916273 | Zbl 1010.11039

[17] Waldschmidt, Michel Integer values entire functions on Cartesian products, Number theory in progress, Vol. 1 (Zakopane-Koscieliko, 1997) (553–576, de Gruyter, Berlin, 1999)

[18] Waldschmidt, Michel Propriétés arithmétiques de fonctions de plusieurs variables. III, Sémin. P. Lelong - H. Skoda, Analyse, Années 1978/79, Lect. Notes Math. 822, 332-356 (1980) (1980) | MR 599036 | Zbl 0444.10028

[19] Waldschmidt, Michel Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften, Berlin Tome 326 (2000) | MR 1756786 | Zbl 0944.11024

[20] Waldschmidt, Michel Algebraic values of analytic functions, J. Comput. Appl. Math., Tome 160 (2003), pp. 323-333 | Article | MR 2022624 | Zbl 1062.11049

[21] Wilkie, A. J. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc., Tome 9 (1996), pp. 1051-1094 | Article | MR 1398816 | Zbl 0892.03013

[22] Wilkie, A. J. A theorem of the complement and some new o-minimal structures, Selecta Math. N.S., Tome 5 (1999), pp. 397-421 | Article | MR 1740677 | Zbl 0948.03037