L'accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 187-1 Solutions entières de l'équation Ym = f(X). par DIMITRIOS POULAKIS Résumé 2014 Soit K un corps de nombres. Dans ce travail nous calculons des majorants effectifs pour la taille des solutions en entiers algébriques de K des équations, Y2 = f(X), où f(X) ~ K[X] a au moins trois racines d'ordre impair, et Ym = f(X) où m ~ 3 et f(X) ~ K[X] a au moins deux racines d'ordre premier à m. On améliore ainsi les estimations connues ([2],[9]) pour les solutions de ces équations en entiers algébriques de K. Abstract 2014 Let K be a number field. In this work we give effective upper bounds for the size of solutions in algebraic integers of K, of equations Y2 = f(X), where f(X) ~ K[X] has at least three roots of odd order, and Ym= f(X) where f(X) ~ K[X] has at least two roots of order prime to m. We thus improve the known estimations ([2],[9]) for the solutions of these
Let F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on C over K. Assume that there is a unit
We prove diophantine inequalities involving various distances between two distinct algebraic points of an algebraic curve. These estimates may be viewed as extensions of classical Liouville's inequality. Our approach is based on a transcendental construction using algebraic functions. Next we apply our results to Hilbert's irreducibility Theorem and to some classes of diophantine equations in the circle of Runge's method. r
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