Geodesic
A generalization of the Bombieri–Pila determinant method
Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 63-77 Cet article a éte moissonné depuis la source Math-Net.Ru

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The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper, we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown's “Theorem 14” by real-analytic considerations alone. Bibl. 11 titles. 
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O. Marmon. A generalization of the Bombieri–Pila determinant method. Zapiski Nauchnykh Seminarov POMI, Studies in number theory. Part 10, Tome 377 (2010), pp. 63-77. http://geodesic.mathdoc.fr/item/ZNSL_2010_377_a9/

[1] J. Bochnak, M. Coste, M.-F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 12, Springer-Verlag, Berlin, 1987 | MR | Zbl

[2] E. Bombieri, J. Pila, “The number of integral points on arcs and ovals”, Duke Math. J., 59:2 (1989), 337–357 | DOI | MR | Zbl

[3] N. Broberg, “A note on a paper by R. Heath-Brown: ‘The density of rational points on curves and surfaces’ ”, J. reine angew. Math., 571 (2004), 159–178 | DOI | MR | Zbl

[4] D. Burguet, “A proof of Yomdin–Gromov's algebraic lemma”, Israel J. Math., 168 (2008), 291–316 | DOI | MR | Zbl

[5] D. Cox, J. Little, D. O'Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992 | DOI | MR | Zbl

[6] M. Gromov, “Entropy, homology, and semi-algebraic geometry”, Astérisque, 145–146:5 (1987), 225–240 | MR | Zbl

[7] D. R. Heath-Brown, “The density of rational points on curves and surfaces”, Ann. Math. (2), 155:2 (2002), 553–595 | DOI | MR

[8] J. Pila, A. J. Wilkie, “The rational points of a definable set”, Duke Math. J., 133:3 (2006), 591–616 | DOI | MR | Zbl

[9] J. Pila, “Integer points on the dilation of a subanalytic surface”, Q. J. Math., 55:2 (2004), 207–223 | DOI | MR | Zbl

[10] P. Salberger, “On the density of rational and integral points on algebraic varieties”, J. reine angew. Math., 606 (2007), 123–147 | DOI | MR | Zbl

[11] Y. Yomdin, “$C^k$-resolution of semi-algebraic mappings”, Addendum to: “Volume growth and entropy”, Israel J. Math., 57:3 (1987), 301–317 | DOI | MR | Zbl