Bounds for rational points on algebraic curves, optimal in the degree, and dimension growth

Binyamini, Gal; Cluckers, Raf; Novikov, Dmitry

Type de document :

Article dans une revue scientifique: Article original

DOI :

10.1093/imrn/rnae034

URL permanente :

http://hdl.handle.net/20.500.12210/121476

Titre :

Bounds for rational points on algebraic curves, optimal in the degree, and dimension growth

Auteur(s) :

Binyamini, Gal [Auteur]
Weizmann Institute of Science [Rehovot, Israël]
Department of Mathematics [Rehovot]
Cluckers, Raf [Auteur]

Université de Lille
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Novikov, Dmitry [Auteur]
Weizmann Institute of Science [Rehovot, Israël]
Department of Mathematics [Rehovot]

Titre de la revue :

International Mathematics Research Notices

Pagination :

9256-9265

Éditeur :

Oxford University Press (OUP)

Date de publication :

2024

ISSN :

1073-7928

Discipline(s) HAL :

Mathématiques [math]

Résumé en anglais : [en]

Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish ...
Lire la suite >Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence on $d$, by showing the upper bound $C d^2 H^{2/d} (\log H)^\kappa$ with some absolute constants $C$ and $\kappa$. This bound is optimal with respect to both $d$ and $H$, except for the constants $C$ and $\kappa$. This answers a question raised by Salberger, leading to a simplified proof of his results on the uniform dimension growth conjectures of Heath-Brown and Serre, and where at the same time we replace the $H^\epsilon$ factor by a power of $\log H$. The main strength of our approach comes from the combination of a new, efficient form of smooth parametrizations of algebraic curves with a century-old criterion of P\'olya, which allows us to save one extra power of $d$ compared with the standard approach using B\'ezout's theorem.Lire moins >

Langue :

Anglais

Comité de lecture :

Oui

Audience :

Internationale

Vulgarisation :

Non

Projet ANR :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :