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Helfgott, Harald
Power-free values, large deviations, and integer points on irrational curves. Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), p. 433-472
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Résumé

Soit $f\in \mathbb{Z}[x]$ 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 $f(p)$ 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.

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