The density of rational points on a pfaff curve
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 3, p. 635-645

This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.

Cet article est concerné par la densité de points rationnels sur le graphe d’une fonction pfaffienne non-algébrique.

@article{AFST_2007_6_16_3_635_0,
     author = {Pila, Jonathan},
     title = {The density of rational points on a pfaff curve},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {3},
     year = {2007},
     pages = {635-645},
     doi = {10.5802/afst.1162},
     mrnumber = {2379055},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_3_635_0}
}
Pila, Jonathan. The density of rational points on a pfaff curve. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 3, pp. 635-645. doi : 10.5802/afst.1162. http://www.numdam.org/item/AFST_2007_6_16_3_635_0/

[1] Bombieri (E) and Pila (J.).— The number of integral points on arcs and ovals, Duke Math. J. 59, p. 337-357 (1989). | MR 1016893 | Zbl 0718.11048

[2] van den Dries (L.).— Tame topology and o-minimal structures, LMS Lecture Note Series 248, CUP, Cambridge, (1998). | Zbl 0953.03045

[3] Gabrielov (A.) and Vorobjov (N.).— Complexity of computations with pfaffian and noetherian functions, in Normal Forms, Bifurcations and Finiteness problems in Differential Equations, Kluwer, (2004). | MR 2083248

[4] Gwozdziewicz (J.), Kurdyka (K.), Parusinski (A.).— On the number of solutions of an algebraic equation on the curve y=e x +sinx,x>0, and a consequence for o-minimal structures, Proc. Amer. Math. Soc. 127, p. 1057-1064 (1999). | MR 1476134 | Zbl 0916.03026

[5] Khovanskii (A. G.).— Fewnomials, Translations of Mathematical Monographs 88, AMS, Providence, (1991). | MR 1108621 | Zbl 0728.12002

[6] Pila (J.).— Integer points on the dilation of a subanalytic surface, Quart. J. Math. 55, p. 207-223 (2004). | MR 2068319 | Zbl 1111.32004

[7] Pila (J.).— Rational points on a subanalytic surface, Ann. Inst. Fourier 55, p. 1501-1516 (2005). | Numdam | MR 2172272 | Zbl 02210717

[8] Pila (J.).— Note on the rational points of a pfaff curve, Proc. Edin. Math. Soc., 49 (2006), 391-397. | MR 2243794 | Zbl 1097.11037

[9] Pila (J.).— Mild parameterization and the rational points of a pfaff curve, Commentari Mathematici Universitatis Sancti Pauli, 55 (2006), 1-8. | MR 2251995 | Zbl 05120444

[10] Pila (J.) and Wilkie (A. J.).— The rational points of a definable set, Duke Math. J., 133 (2006), 591-616. | MR 2228464 | Zbl 05043321

[11] Pólya (G.).— On the zeros of the derivative of a function and its analytic character, Bull. Amer. Math. Soc. 49, 178-191 (1943). Also Collected Papers: Volume II, MIT Press, Cambridge Mass., p. 394-407 (1974). | MR 7781 | Zbl 0061.11510

[12] Waldschmidt (M.).— Diophantine approximation on linear algebraic groups, Grund. Math. Wissen. 326, Springer, Berlin, (2000). | MR 1756786 | Zbl 0944.11024

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