Polynomial bound for the number of rational points on curves
type: Item
instance of: scholarly article
title: Polynomial bound for the number of rational points on curves
zbMATH Open document ID: 1247.14022
author: Gaël Rémond
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2012-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/219697
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G50
zbMATH DE Number: 6075042
zbMATH Keywords: Curves over number fields
zbMATH Keywords: number of rational points
MaRDI profile type: Publication
OpenAlex ID: W2317143317
cites work: Minorations des hauteurs normalisées des sous-variétés de variétés abeliennes. II. (Lower bounds for normalized heights of subvarieties of Abelian varieties. II.)
cites work: Counting in a conjecture of Lang
cites work: Nombre de points rationnels des courbes
DOI: 10.5802/JTNB.759
review text: 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 NEWLINE\[NEWLINE \text{exp}\left(2^{3^{D^2}}[K:{\mathbb Q}]^4 \max (h_\infty(F), \log|\Delta|, 1)\right), NEWLINE\]NEWLINE where \(h_\infty(F)\) denotes the logarithm of the maximum archimedean absolute value of the coefficients of \(F\).NEWLINENEWLINEThe 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.
description: scientific article; zbMATH DE number 6075042