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

Boutabaa, Abdelbaki; Escassut, Alain

doi:10.5802/aif.1771

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.

Résumé
Abstract

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.

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 $y^{' m} = F (y)$ , 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 $f (x) = a + λ (x - α)^{\frac{m}{m - d}}$ , with $λ^{m - d} (\frac{m}{m - d})^{m} = A$ .

EuDML MR Zbl

DOI : 10.5802/aif.1771

@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},
     pages = {751--766},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {3},
     year = {2000},
     doi = {10.5802/aif.1771},
     mrnumber = {2002a:30073},
     zbl = {1063.30043},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1771/}
}

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
SP  - 751
EP  - 766
VL  - 50
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1771/
DO  - 10.5802/aif.1771
LA  - en
ID  - AIF_2000__50_3_751_0
ER  -

%0 Journal Article
%A Boutabaa, Abdelbaki
%A Escassut, Alain
%T Applications of the $p$-adic Nevanlinna theory to functional equations
%J Annales de l'Institut Fourier
%D 2000
%P 751-766
%V 50
%N 3
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1771/
%R 10.5802/aif.1771
%G en
%F 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, Tome 50 (2000) no. 3, pp. 751-766. doi : 10.5802/aif.1771. http://www.numdam.org/articles/10.5802/aif.1771/

Bibliographie
Cité par

[1] W. Berkovich, Spectral Theory and Analytic Geometry over Non-archimedean Fields, AMS Surveys and Monographs, 33 (1990). | MR | Zbl

[2] A. Boutabaa, Théorie de Nevanlinna p-adique, Manuscripta Mathematica, 67 (1990), 251-269. | MR | Zbl

[3] A. Boutabaa, A. Escassut, An Improvement of the p-adic Nevanlinna Theory and Application to Meromorphic Functions, Lecture Notes in Pure and Applied Mathematics n° 207 (Marcel Dekker). | MR | Zbl

[4] A. Boutabaa, On some p-adic functional equations, Lecture Notes in Pure and Applied Mathematics (Marcel Dekker), 192 (1997), 49-59. | MR | Zbl

[5] A. Boutabaa, A. Escassut, and L. Haddad, On uniqueness of p-adic entire functions, Indagationes Mathematicae, 8 (1997), 145-155. | MR | Zbl

[6] A. Boutabaa, A. Escassut, Urs and ursim for p-adic unbounded analytic functions inside a disk, (preprint).

[7] A. Boutabaa, A. Escassut, Property f— (S) = g— (S) for p-adic entire and meromorphic functions, to appear in Rendiconti del Circolo Matematico di Palermo.

[8] W. Cherry, Non-archimedean analytic curves in Abelian varieties, Math. Ann., 300 (1994), 393-404. | MR | Zbl

[9] A. Escassut, Analytic Elements in p-adic Analysis, World Scientific Publishing Co. Pte. Ltd., Singapore, 1995. | MR | Zbl

[10] A. Escassut, L. Haddad, and R. Vidal, Urs, ursim, and non-urs, Journal of Number Theory, 75 (1999), 133-144. | MR | Zbl

[11] F. Gross, On the equation fn + gn = 1, Bull. Amer. Math. Soc., 72 (1966), 86-88. | MR | Zbl

[12] I. Kaplansky, An Introduction to Differential Algebra, Actualités Scientifiques et Industrielles 1251, Hermann, Paris (1957). | MR | Zbl

[13] R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Gauthiers-Villars, Paris, 1929. | JFM

[14] E. Picard, Traité d'analyse II, Gauthier-Villars, Paris, 1925.

Cité par Sources :