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.
Accepted:
DOI: 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}, pages = {489--514}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {2}, year = {2010}, doi = {10.5802/aif.2530}, zbl = {1210.11074}, mrnumber = {2667784}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2530/} }
TY - JOUR TI - Counting rational points on a certain exponential-algebraic surface JO - Annales de l'Institut Fourier PY - 2010 DA - 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/ UR - https://zbmath.org/?q=an%3A1210.11074 UR - https://www.ams.org/mathscinet-getitem?mr=2667784 UR - https://doi.org/10.5802/aif.2530 DO - 10.5802/aif.2530 LA - en ID - AIF_2010__60_2_489_0 ER -
Pila, Jonathan. 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/articles/10.5802/aif.2530/
[1] Algebraic values of meromorphic maps, Inventiones, Tome 10 (1970), pp. 267-287 | Article | MR: 306201 | Zbl: 0214.33702
[2] Heights in Diophantine geometry, New Mathematical Monographs 4. Cambridge: Cambridge University Press. xvi, 652 p., 2007 | MR: 2216774 | Zbl: 1115.11034
[3] 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] Some cases of Wilkie’s conjecture (2009) (working paper)
[5] 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] Geometric categories and o-minimal structures, Duke Math. J., Tome 84 (1996) no. 2, pp. 497-540 | Article | MR: 1404337 | Zbl: 0889.03025
[7] Complexity of computations with Pfaffian and Noetherian functions, Normal Forms, Bifurcations and Finiteness problems in Differential Equations, Kluwer, 2004 | MR: 2083248
[8] Algebraic values of meromorphic functions, Topology, Tome 3 (1965), pp. 183-191 | Article | MR: 190092 | Zbl: 0133.13804
[9] Introduction to transcendental numbers, Addison-Wesley, Reading Mass, 1966 | MR: 214547 | Zbl: 0144.04101
[10] 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] 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] 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] The density of rational points on a pfaff curve, Ann. Fac. Sci. Toulouse, Tome 16 (2007), pp. 635-645 | Article | Numdam | MR: 2379055
[14] On the algebraic points of a definable set, Selecta Math. N.S., Tome 15 (2009), pp. 151-170 | Article | MR: 2511202
[15] The rational points of a definable set, Duke Math. J., Tome 133 (2006), pp. 591-616 | Article | MR: 2228464 | Zbl: pre05043321
[16] Interpolation formulas and auxiliary functions, J. Number Theory, Tome 94 (2002), pp. 248-285 | Article | MR: 1916273 | Zbl: 1010.11039
[17] Integer values entire functions on Cartesian products, Number theory in progress, Vol. 1 (Zakopane-Koscieliko, 1997) (553–576, de Gruyter, Berlin, 1999)
[18] 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] Diophantine approximation on linear algebraic groups Tome 326, Grundlehren der Mathematischen Wissenschaften, Berlin, 2000 | MR: 1756786 | Zbl: 0944.11024
[20] Algebraic values of analytic functions, J. Comput. Appl. Math., Tome 160 (2003), pp. 323-333 | Article | MR: 2022624 | Zbl: 1062.11049
[21] 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] 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
Cited by Sources: