no. 2
Counting rational points on a certain exponential-algebraic surface
[Comptage des points rationnels sur une surface exponentielle-algébrique particulière]
Pila, Jonathan1
1 University of Bristol School of Mathematics Bristol, BS8 1TW (United Kingdom)
Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 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 : 10.5802/aif.2530
Classification : 11G99, 03C64
Keywords: O-minimal structure, rational points, transcendental numbers
Mots-clés : structure o-minimale, points rationnels, nombres transcendants

Pila, Jonathan 1

1 University of Bristol School of Mathematics Bristol, BS8 1TW (United Kingdom) 
@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},
     pages = {489--514},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {2},
     year = {2010},
     doi = {10.5802/aif.2530},
     mrnumber = {2667784},
     zbl = {1210.11074},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2530/}
}
TY  - JOUR
AU  - Pila, Jonathan
TI  - Counting rational points on a certain exponential-algebraic surface
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 489
EP  - 514
VL  - 60
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2530/
DO  - 10.5802/aif.2530
LA  - en
ID  - AIF_2010__60_2_489_0
ER  - 
%0 Journal Article
%A Pila, Jonathan
%T Counting rational points on a certain exponential-algebraic surface
%J Annales de l'Institut Fourier
%D 2010
%P 489-514
%V 60
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2530/
%R 10.5802/aif.2530
%G en
%F AIF_2010__60_2_489_0
Pila, Jonathan. Counting rational points on a certain exponential-algebraic surface. Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 489-514. doi : 10.5802/aif.2530. https://aif.centre-mersenne.org/articles/10.5802/aif.2530/

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

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

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

[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 | Zbl

[6] van den Dries, Lou; Miller, C. Geometric categories and o-minimal structures, Duke Math. J., Volume 84 (1996) no. 2, pp. 497-540 | DOI | MR | Zbl

[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

[8] LANG, S. Algebraic values of meromorphic functions, Topology, Volume 3 (1965), pp. 183-191 | DOI | MR | Zbl

[9] LANG, S. Introduction to transcendental numbers, Addison-Wesley, Reading Mass, 1966 | MR | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

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

[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., Volume 9 (1996), pp. 1051-1094 | DOI | MR | Zbl

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

Cité par Sources :