Polynomial bound for the number of rational points on curves. (Borne polynomiale pour le nombre de points rationnels des courbes.) (French. English summary) Zbl 1247.14022

Let \(k\) be a number field, with discriminant \(\Delta\). Let \(f\in k[X,Y]\) be a polynomial of degree \(D\). The main theorem of the paper states that the set of affine \(k\)-rational points of \(f(x,y)=0\) is either infinite or bounded by \[ \text{exp}\left(2^{3^{D^2}}[K:{\mathbb Q}]^4 \max (h_\infty(F), \log|\Delta|, 1)\right), \] where \(h_\infty(F)\) denotes the logarithm of the maximum archimedean absolute value of the coefficients of \(F\).
The paper uses a simple estimate of the number of points of \({\mathbb P}^n(\bar{\mathbb Q})\) of degree \(\leq D\) and of height \(\leq H\) and compares with a previous paper of the same author [Proc. Lond. Math. Soc. (3) 101, No. 3, 759–794 (2010; Zbl 1210.11073)], improving the term of the estimate concerning the height but losing (substantially) on the dependence on the degree.


14G05 Rational points
11G50 Heights


Zbl 1210.11073
Full Text: DOI EuDML


[1] S. David et P. Philippon, Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes. Comment. Math. Helv. 77 (2002), 639-700. · Zbl 1030.11026
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[3] G. Rémond, Nombre de points rationnels des courbes. Proc. London Math. Soc. 101 (2010), 759-794. · Zbl 1210.11073
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