Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On uniqueness of $p$-adic meromorphic functions

Authors: Abdelbaki Boutabaa and Alain Escassut
Journal: Proc. Amer. Math. Soc. 126 (1998), 2557-2568
MSC (1991): Primary 11Q25
MathSciNet review: 1468183
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $K$ be a complete ultrametric algebraically closed field of characteristic zero, and let ${\mathcal{M}} (K)$ be the field of meromorphic functions in $K$. For all set $S$ in $ K$ and for all $f\in {\mathcal{M}}(K)$ we denote by $\displaystyle E(f,S)$ the subset of $K {\times } {\mathbb{N}}^{*}$: ${\bigcup _{ a\in S}}\{(z,q)\in K {\times } \mathbb{N}^{*} \vert \ z$ zero of order $ q \ \text{ of} \ f(z)-a\}.$ After studying unique range sets for entire functions in $K$ in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every $n\geq 5$, there exist sets $S$ of $n$ elements in $K$ such that, if $f,\ g\in {\mathcal{M}} (K)$ have the same poles (counting multiplicities), and satisfy $E(f,S)=E(g,S)$, then $f=g$. We show how to construct such sets.

References [Enhancements On Off] (What's this?)

  • 1. W. W. Adams and E. G. Straus, Non-archimedian analytic functions taking the same values at the same points, Illinois J. Math. 15 (1971), 418–424. MR 0277771
  • 2. Yvette Amice, Les nombres 𝑝-adiques, Presses Universitaires de France, Paris, 1975 (French). Préface de Ch. Pisot; Collection SUP: Le Mathématicien, No. 14. MR 0447195
  • 3. Abdelbaki Boutabaa, Théorie de Nevanlinna 𝑝-adique, Manuscripta Math. 67 (1990), no. 3, 251–269 (French, with English summary). MR 1046988,
  • 4. Boutabaa, A. Escassut, A. and Haddad, L. On uniqueness of p-adic entire functions. To appear in Indagationes Mathematicae (1997).
  • 5. Boutabaa, A. and Escassut, A. Uniqueness of p-adic meromorphic functions. Comptes Rendus de l'Académie des Sciences, Paris, t; 325, Serie I, p. 571-575, 1997. CMP 98:02
  • 6. W. Cherry and C.-C. Yang Uniqueness of non-Archimedean entire functions sharing sets of values counting multiplicities, to appear in the Proceedings of the AMS.
  • 7. Alain Escassut, Algèbres d’éléments analytiques en analyse non archimédienne, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 339–351 (French). MR 0374471
  • 8. Alain Escassut, Éléments analytiques et filtres percés sur un ensemble infraconnexe, Ann. Mat. Pura Appl. (4) 110 (1976), 335–352 (French, with English summary). MR 0425175,
  • 9. Alain Escassut, Analytic elements in 𝑝-adic analysis, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1370442
  • 10. Frank, G. and Reinders, M. A unique Range set for meromorphic functions with 11 eleven elements, to appear in Complex Variable.
  • 11. G. Garandel, Les semi-normes multiplicatives sur les algèbres d’éléments analytiques au sens de Krasner, Nederl. Akad. Wetensch. Proc. Ser. A 78=Indag. Math. 37 (1975), no. 4, 327–341 (French). MR 0390286
  • 12. Gross, F. Factorization of meromorphic functions and some open problems. Lecture Notes in pure and Applied Math. 78, 51-67 (1982).
  • 13. Fred Gross and Chung Chun Yang, On preimage and range sets of meromorphic functions, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 1, 17–20. MR 649056
  • 14. Marc Krasner, Prolongement analytique uniforme et multiforme dans les corps valués complets, Les Tendances Géom. en Algèbre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique, Paris, 1966, pp. 97–141 (French). MR 0204404
  • 15. Hong Xun Yi, On a question of Gross, Sci. China Ser. A 38 (1995), no. 1, 8–16. MR 1335194
  • 16. E. Mues and M. Reinders, Meromorphic functions sharing one value and unique range sets, Kodai Math. J. 18 (1995), no. 3, 515–522. MR 1362926,
  • 17. Ping Li and Chung-Chun Yang, On the unique range set of meromorphic functions, Proc. Amer. Math. Soc. 124 (1996), no. 1, 177–185. MR 1291784,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11Q25

Retrieve articles in all journals with MSC (1991): 11Q25

Additional Information

Abdelbaki Boutabaa
Affiliation: Laboratoire de Mathématiques Pures, Université Blaise Pascal, (Clermont-Ferrand), Les Cézeaux, 63177 Aubiere Cedex, France

Alain Escassut
Affiliation: Laboratoire de Mathématiques Pures, Université Blaise Pascal, (Clermont-Ferrand), Les Cézeaux, 63177 Aubiere Cedex, France

Received by editor(s): October 22, 1996
Received by editor(s) in revised form: December 10, 1996, and January 31, 1997
Communicated by: William W. Adams
Article copyright: © Copyright 1998 American Mathematical Society