Rational points on a subanalytic surface
[Points rationnels d'une surface sous-analytique]
Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1501-1516.

Soit X n une surface sous-analytique compacte. Cet article démontre qu’en un sens convenable, il y a très peu de points rationnels de X qui ne se trouvent pas sur une courbe semi-algébrique connexe contenue dans X.

Let X n be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of X that do not lie on some connected semialgebraic curve contained in X.

DOI : 10.5802/aif.2131
Classification : 11D99, 11J99
Keywords: Subanalytic set, rational point
Mot clés : ensemble sous-analytique, point rationnel
Pila, Jonathan 1

1 McGill University, department of mathematics and statistics, Burnside Hall, 805 Sherbrooke Street West, Montreal, Quebec, H3A 2K6 (Canada), University of Oxford, mathematical institute, 24-29 St Giles, Oxford OX1 3LB (Grande-Bretagne)
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Pila, Jonathan. Rational points on a subanalytic surface. Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1501-1516. doi : 10.5802/aif.2131. http://archive.numdam.org/articles/10.5802/aif.2131/

[1] E. Bierstone; P.D. Milman Semianalytic and subanalytic sets, Pub. Math. I.H.E.S, Volume 67 (1988), pp. 5-42 | Numdam | MR | Zbl

[2] E. Bombieri email, 9 March (2003)

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

[4] D. R. Heath-Brown The density of rational points on curves and surfaces, Ann. Math., Volume 155 (2002), pp. 553-595 | DOI | MR | Zbl

[5] M. Hindry; J. H. Silverman Diophantine geometry: an introduction, Graduate Texts in Mathematics, 201, Springer, New York, 2000 | MR | Zbl

[6] V. Jarnik Über die Gitterpunkte auf konvexen Curven, Math. Z., Volume 24 (1926), pp. 500-518 | DOI | JFM | MR

[7] S. Lang Number theory III: diophantine geometry, Encyclopedia of Mathematical Sciences, 60, Springer, Berlin, 1991 | MR | Zbl

[8] J. Pila Geometric postulation of a smooth function and the number of rational points, Duke Math. J., Volume 63 (1991), pp. 449-463 | MR | Zbl

[9] J. Pila Density of integer points on plane algebraic curves, International Mathematics Research Notices (1996), pp. 903-912 | MR | Zbl

[10] J. Pila Integer points on the dilation of a subanalytic surface, Quart. J. Math., Volume 55 (2004), pp. 207-223 | DOI | MR | Zbl

[11] J. Pila Note on the rational points of a pfaff curve (submitted)

[12] W. M. Schmidt Integer points on curves and surfaces, Monatsh. Math., Volume 99 (1985), pp. 45-72 | DOI | MR | Zbl

[13] H. P. F. Swinnerton-Dyer The number of lattice points on a convex curve, J. Number Theory, Volume 6 (1974), pp. 128-135 | DOI | MR | Zbl

[14] A. Wilkie Diophantine properties of sets definable in o-minimal structures, J. Symb. Logic, Volume 69 (2004), pp. 851-861 | DOI | MR | Zbl

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