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

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.

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 fm+gn=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 ym=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)mm-d, with λm-d(mm-d)m=A.

@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/

[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.

  • Boughaba, Houda; Zerzaihi, Tahar On the growth order of meromorphic solutions of some ultrametric q-difference equations, Lobachevskii Journal of Mathematics, Volume 44 (2023) no. 4, pp. 1280-1288 | DOI:10.1134/s1995080223040066 | Zbl:1521.30055
  • Bouternikh, Salih; Zerzaihi, Tahar On some properties of ultrametric meromorphic solutions of Malmquist type, Russian Mathematics, Volume 66 (2022) no. 8, pp. 19-26 | DOI:10.3103/s1066369x22080023 | Zbl:1509.30031
  • Escassut, A.; Riquelme, J.-l. Applications of branched values to p-adic functional equations on analytic functions, p-Adic Numbers, Ultrametric Analysis, and Applications, Volume 6 (2014) no. 3, pp. 188-194 | DOI:10.1134/s2070046614030029 | Zbl:1309.30040
  • Boudjerida, Nadjet; Boutabaa, Abdelbaki; Medjerab, Samia On some ultrametric q-difference equations, Bulletin des Sciences Mathématiques, Volume 137 (2013) no. 2, p. 177 | DOI:10.1016/j.bulsci.2010.05.002
  • Boutabaa, Abdelbaki About the p-adic Yosida equation inside a disk, Indagationes Mathematicae. New Series, Volume 20 (2009) no. 3, pp. 397-413 | DOI:10.1016/s0019-3577(10)00005-4 | Zbl:1203.30047
  • Ojeda, Jacqueline HAYMAN’S CONJECTURE IN A p-ADIC FIELD, Taiwanese Journal of Mathematics, Volume 12 (2008) no. 9 | DOI:10.11650/twjm/1500405180
  • Escassut, Alain; Yang, Chung-Chun The functional equation P(f)=Q(g) in a p-adic field., Journal of Number Theory, Volume 105 (2004) no. 2, pp. 344-360 | DOI:10.1016/j.jnt.2003.11.005 | Zbl:1054.30043
  • Boutabaa, Abdelbaki; Escassut, Alain Nevanlinna Theory in Characteristic P and Applications, Analysis and Applications — ISAAC 2001, Volume 10 (2003), p. 97 | DOI:10.1007/978-1-4757-3741-7_7
  • Vidaux, X. An analogue of Hilbert's 10th problem for fields of meromorphic functions over non-Archimedean valued fields, Journal of Number Theory, Volume 101 (2003) no. 1, pp. 48-73 | DOI:10.1016/s0022-314x(03)00016-7 | Zbl:1028.03013
  • Boutabaa, Abdelbaki A note on p-adic linear differential equations, Journal of Number Theory, Volume 87 (2001) no. 2, pp. 301-305 | DOI:10.1006/jnth.2000.2602 | Zbl:0993.12004

Cité par 10 documents. Sources : Crossref, zbMATH