Applications of the $p$-adic Nevanlinna theory to functional equations
Annales de l'Institut Fourier, Volume 50 (2000) no. 3, p. 751-766

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 $f,\phantom{\rule{4pt}{0ex}}g$ 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 $A\left(y-a{\right)}^{d}$ where $m-d$ divides $m$, and then the solutions are the functions of the form $f\left(x\right)=a+\lambda \left(x-\alpha {\right)}^{\frac{m}{m-d}}$, with ${\lambda }^{m-d}\left(\frac{m}{m-d}{\right)}^{m}=A$.

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.

@article{AIF_2000__50_3_751_0,
author = {Boutabaa, Abdelbaki and Escassut, Alain},
title = {Applications of the $p$-adic Nevanlinna theory to functional equations},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {50},
number = {3},
year = {2000},
pages = {751-766},
doi = {10.5802/aif.1771},
zbl = {1063.30043},
zbl = {01478802},
mrnumber = {2002a:30073},
language = {en},
url = {http://www.numdam.org/item/AIF_2000__50_3_751_0}
}

Boutabaa, Abdelbaki; Escassut, Alain. Applications of the $p$-adic Nevanlinna theory to functional equations. Annales de l'Institut Fourier, Volume 50 (2000) no. 3, pp. 751-766. doi : 10.5802/aif.1771. http://www.numdam.org/item/AIF_2000__50_3_751_0/

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