Applications of the $p$-adic Nevanlinna theory to functional equations

Boutabaa, Abdelbaki ; Escassut, Alain

Annales de l'Institut Fourier, Tome 50 (2000), p. 751-766 / Harvested from Numdam

Résumé

Soit $K$ un corps ultramétrique complet algébriquement clos de caractéristique nulle. On applique la théorie de Nevanlinna $p$-adique aux équations de la forme $g= R\circ f$, où $R\in K(x)$, et $f, g$ sont des fonctions méromorphes dans $K$ ou dans un disque ouvert, ainsi qu’à l’équation de Yoshida.

Abstract

Let $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the $p$-adic Nevanlinna theory to functional equations of the form $g= R\circ f$, where $R\in K(x)$, $f, g$ are meromorphic functions in $K$, or in an “open disk”, $g$ satisfying conditions on the order of its zeros and poles. In various cases we show that $f$ and $g$ must be constant when they are meromorphic in all $K$, or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular, we have an easy way to obtain again Picard-Berkovich’s theorem for curves of genus $1$ and $2$. These results apply to equations $f^m+g^n=1$, when $f,\ g$ are meromorphic functions, or entire functions in $K$ or analytic functions in an “open disk”. We finally apply the method to Yoshida’s equation $y^{\prime m}= F(y)$, when $F\in K(X)$, and we describe the only case where solutions exist: $F$ must be a polynomial of the form $A(y-a)^d$ where $m-d $ divides $m$, and then the solutions are the functions of the form $f(x)=a+\lambda (x-\alpha )^{m\over m-d}$, with $\lambda ^{m-d}({m\over m-d})^m=A$.

Détail

Publié le : 2000-01-01

MR 2002a:30073 | Zbl 01478802 | Zbl 1063.30043

DOI : https://doi.org/10.5802/aif.1771

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