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

Helfgott, Harald A.

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

Soit $f\in \mathbb{Z}\left[x\right]$ un polynôme de degré $d\ge 3$ sans racines de multiplicité $d$ ou $(d-1)$. Erdős a conjecturé que si $f$ satisfait les conditions locales nécessaires alors $f\left(p\right)$ est sans facteurs puissances ${(d-1)}^{\text{èmes}}$ 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 $f\in \mathbb{Z}\left[x\right]$ be a polynomial of degree $d\ge 3$ without roots of multiplicity $d$ or $(d-1)$. Erdős conjectured that, if $f$ satisfies the necessary local conditions, then $f\left(p\right)$ 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.

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