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

Boutabaa, Abdelbaki; Escassut, Alain

Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 751-766.

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 $R\in K\left(x\right)$, et $f,g$ sont des fonctions méromorphes dans $K$ ou dans un disque ouvert, ainsi qu’à l’équation de Yoshida.

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 $R\in K\left(x\right)$, $f,g$ 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 $y^{\prime m}=F\left(y\right)$, when $F\in K\left(X\right)$, 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 $f\left(x\right)=a+\lambda (x-\alpha {)}^{\frac{m}{m-d}}$, with ${\lambda }^{m-d}(\frac{m}{m-d}{)}^m=A$.

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