Nouvelles méthodes pour minorer des combinaisons linéaires de logarithmes de nombres algébriques

Waldschmidt, Michel

Séminaire de théorie des nombres de Bordeaux, Série 2, Tome 3 (1991) no. 1, pp. 129-185.

Depuis un peu plus de vingt ans, la recherche de minorations de combinaisons linéaires de logarithmes de nombres algébriques avec des coefficients algébriques a fait l'objet de nombreux travaux. Dès que le nombre de logarithmes dépasse 2, toutes les démonstrations utilisées jusqu'à présent reposaient sur la méthode de Baker. Nous proposons ici d'autres méthodes.

Up to now Baker’s method was the only one to yield lower bounds for linear forms in logarithms of algebraic numbers, at least when the number of logarithms is bigger than $2$. We propose here other methods. While Baker’s work is a generalization of Gel’fond’s solution of Hilbert’s problem, our approach is based on Schneider’s solution of this problem. This first paper is devoted to the “dual” of a proof due to N. Hirata of a lower bound for linear forms in a commutative algebraic group (here we consider only the usual logarithms). In a subsequent paper we shall develop the “dual” of Baker’s method.

Mots clés : formes linéaires de logarithmes, transcendance, approximation diophantienne, méthodes de Schneider, Gel'fond, Baker

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