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

Abdelbaki Boutabaa; Alain Escassut

Annales de l'institut Fourier (2000)

- Volume: 50, Issue: 3, page 751-766
- ISSN: 0373-0956

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topBoutabaa, 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 -

## References

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