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

Boutabaa, Abdelbaki, and Escassut, Alain. "Applications of the $p$-adic Nevanlinna theory to functional equations." Annales de l'institut Fourier 50.3 (2000): 751-766. <http://eudml.org/doc/75436>.

@article{Boutabaa2000,
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 $\lambda ^\{m-d\}(\{m\over m-d\})^m=A$.},
author = {Boutabaa, Abdelbaki, Escassut, Alain},
journal = {Annales de l'institut Fourier},
keywords = {Picard-Berkovich's theorem},
language = {eng},
number = {3},
pages = {751-766},
publisher = {Association des Annales de l'Institut Fourier},
title = {Applications of the $p$-adic Nevanlinna theory to functional equations},
url = {http://eudml.org/doc/75436},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Boutabaa, Abdelbaki
AU - Escassut, Alain
TI - Applications of the $p$-adic Nevanlinna theory to functional equations
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 3
SP - 751
EP - 766
AB - 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 $\lambda ^{m-d}({m\over m-d})^m=A$.
LA - eng
KW - Picard-Berkovich's theorem
UR - http://eudml.org/doc/75436
ER -