Power-free values, large deviations, and integer points on irrational curves

Harald A. Helfgott

doi:10.5802/jtnb.596

Harald A. Helfgott ¹

¹ Département de mathématiques et de statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 433-472.

Résumé
Abstract

Soit $f \in ℤ [x]$ un polynôme de degré $d \geq 3$ sans racines de multiplicité $d$ ou $(d - 1)$ . Erdős a conjecturé que si $f$ satisfait les conditions locales nécessaires alors $f (p)$ est sans facteurs puissances ${(d - 1)}^{èmes}$ pour une infinité de nombres premiers $p$ . On prouve cela pour toutes les fonctions $f$ dont l’entropie est assez grande.

On utilise dans la preuve un principe de répulsion pour les points entiers sur les courbes de genre positif et un analogue arithmétique du théorème de Sanov issu de la théorie des grandes déviations.

Let $f \in ℤ [x]$ be a polynomial of degree $d \geq 3$ without roots of multiplicity $d$ or $(d - 1)$ . Erdős conjectured that, if $f$ satisfies the necessary local conditions, then $f (p)$ is free of $(d - 1)$ th powers for infinitely many primes $p$ . This is proved here for all $f$ with sufficiently high entropy.

The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations.

MR

DOI : 10.5802/jtnb.596

Affiliations des auteurs :

Harald A. Helfgott ¹

¹ Département de mathématiques et de statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada

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Harald A. Helfgott. Power-free values, large deviations, and integer points on irrational curves. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 433-472. doi : 10.5802/jtnb.596. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.596/

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