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=Rf, where RK(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 FK(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) m m-d , with λ m-d (m m-d) 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=Rf, où RK(x), 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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