Power-free values of polynomials and integer points on irrational curves
نویسندگان
چکیده
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’autres fonctions. La preuve utilise d’une part une propriété de repulsion entre les points entiers des courbes de genre positif et d’autre part des arguments probabilistes.
منابع مشابه
Power - free values , large deviations , and integer points on irrational curves par
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 ...
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