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The number of integral points on arcs and ovals

- E. Bombieri, J. Pila
- Mathematics
- 1 October 1989

integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed,… Expand

244 57- PDF

The rational points of a definable set

Let $X\R^n$ be a set that is definable in an o-minimal structure over $R$. This article shows that in a suitable sense, there are very few rational points of $X$ which do not lie on some connected… Expand

175 31- PDF

Rational points in periodic analytic sets and the Manin-Mumford conjecture

- J. Pila, U. Zannier
- Mathematics
- 27 February 2008

We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points… Expand

118 14- PDF

Frobenius maps of Abelian varieties and finding roots of unity in finite fields

- J. Pila
- Mathematics
- 13 January 1990

"If 'twere done when 'tis done, then 'twere well/ It were done quickly."-Macbeth. Abstract. We give a generalization to Abelian varieties over finite fields of the algorithm of Schoof for elliptic… Expand

188 9- PDF

O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$

- J. Pila
- Mathematics
- 1 May 2011

We give an unconditional proof of the Andre-Oort conjecture for arbitrary products of modular curves. We establish two generalizations. The first includes the Manin-Mumford conjecture for arbitrary… Expand

132 9- PDF

O-minimality and certain atypical intersections

- P. Habegger, J. Pila
- Mathematics
- 2 September 2014

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We… Expand

49 9- PDF

Rational Points of Definable Sets and Results of André–Oort–Manin–Mumford type

- J. Pila
- Mathematics
- 2009

We prove some simple special cases, partly new, of results of Andre-Oort-Manin-Mumford type using an extension to algebraic points of bounded degree of a result of Pila-Wilkie on the density of… Expand

31 8- PDF

Density of integral and rational points on varieties

- J. Pila
- Mathematics
- 1995

© Société mathématique de France, 1995, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les… Expand

48 8- PDF

Integer points on the dilation of a subanalytic surface

- J. Pila
- Mathematics
- 1 June 2004

Let � ⊂ R n be a compact subanalytic set of dimension 2 and t 1. This paper gives an upper bound as t →∞ for the number of integer points on the homothetic dilation tofthat do not reside on any… Expand

56 7- PDF

On the algebraic points of a definable set

- J. Pila
- Mathematics
- 8 May 2009

Abstract.This paper studies diophantine properties of sets definable in an o-minimal structure over the real field. The main theorem of the author’s recent paper with A. J. Wilkie is refined, and… Expand

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