Power - free values , large deviations , and integer points on irrational curves par
نویسندگان
چکیده
Résumé. Soit f ∈ Z[x] un polynôme de degré d ≥ 3 sans racines de multiplicité d ou (d− 1). Erdős a conjecturé que si f satisfait les conditions locales necessaires alors f(p) est sans facteurs puissances (d − 1) 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.
منابع مشابه
Power-free values of polynomials and integer points on irrational curves
Résumé. Soit f ∈ Z[x] un polynôme de degré r ≥ 3 sans racines de multiplicité r ou (r − 1). Supposons que f(x) 6≡ 0 mod p admette une solution dans (Z/p) pour tout p. Erdős a conjecturé que f(p) est donc sans facteurs puissances (r − 1)ièmes pour un nombre infini de premiers p. On prouve cela pour toutes les fonctions f dont une racine génère le corps de décomposition, et également pour d’autre...
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