On the integer solutions of exponential equations in function fields

Zannier, Umberto

Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 849-874.

On étudie le nombre de solutions entières d’équations exponentielles à plusieurs variables sur les corps de fonctions. On développe une méthode qui, dans certains cas, permet de remplacer des bornes exponentielles par des bornes polynomiales. Puis, on démontre un résultat de comptage (Thm.1) des points entiers où des termes exponentiels deviennent linéairement dépendants sur le corps des constantes. On fournit plusieurs applications aux équations (Cor. 1) et aux estimations du nombre de valeurs où certaines paires de suites récurrentes linéaires coïncident (Cor. 2). En particulier, on améliore sensiblement (Cor. 3) des bornes récentes pour le nombre de solutions entières $(m,n)$ de l’équation $G_m\left(P\left(X\right)\right)=c_{m,n}{G}_n\left(X\right)$, où $G_n$ est une suite récurrente linéaire de polynômes et $c_{m,n}$ appartient au corps des constantes. Enfin (Cor. 4), on estime le nombre de solutions d’une équation en $S$-unités à deux variables sur un corps de fonctions, en améliorant les bornes connues.

This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal values of certain pairs of recurrence sequences (Cor. 2). In particular we substantially sharpen (Cor. 3) recent bounds for the number of integer solutions $(m,n)$ of $G_m\left(P\left(X\right)\right)=c_{m,n}{G}_n\left(X\right)$, where $G_n$ is a recurrence of polynomials, $P$ is a polynomial and $c_{m,n}$ is a variable constant. Finally, we estimate the number of solutions to an $S$-unit type equation in two variables (Cor. 4), improving on known bounds.

Classification : 11D45, 11D61, 11D99
Keywords: number theory, diophantine equations, function fields
Mot clés : théorie des nombres, équations diophantiennes, corps de fonctions

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