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Proceedings of the American Mathematical Society

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On uniqueness of $p$-adic meromorphic functions
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by Abdelbaki Boutabaa and Alain Escassut PDF
Proc. Amer. Math. Soc. 126 (1998), 2557-2568 Request permission

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
  • 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 277771, DOI 10.1215/ijm/1256052610
  • Yvette Amice, Les nombres $p$-adiques, Collection SUP: “Le Mathématicien”, vol. 14, Presses Universitaires de France, Paris, 1975 (French). Préface de Ch. Pisot. MR 0447195
  • Abdelbaki Boutabaa, Théorie de Nevanlinna $p$-adique, Manuscripta Math. 67 (1990), no. 3, 251–269 (French, with English summary). MR 1046988, DOI 10.1007/BF02568432
  • Boutabaa, A. Escassut, A. and Haddad, L. On uniqueness of p-adic entire functions. To appear in Indagationes Mathematicae (1997).
  • 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.
  • 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.
  • 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, DOI 10.1016/1385-7258(74)90024-9
  • 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 425175, DOI 10.1007/BF02418012
  • Alain Escassut, Analytic elements in $p$-adic analysis, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1370442, DOI 10.1142/9789812831019
  • Frank, G. and Reinders, M. A unique Range set for meromorphic functions with 11 eleven elements, to appear in Complex Variable.
  • 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, DOI 10.1016/1385-7258(75)90004-9
  • Gross, F. Factorization of meromorphic functions and some open problems. Lecture Notes in pure and Applied Math. 78, 51-67 (1982).
  • 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
  • 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 (CNRS), Paris, 1966, pp. 97–141 (French). MR 0204404
  • Hong Xun Yi, On a question of Gross, Sci. China Ser. A 38 (1995), no. 1, 8–16. MR 1335194
  • 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, DOI 10.2996/kmj/1138043489
  • 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, DOI 10.1090/S0002-9939-96-03045-6
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Additional Information
  • Abdelbaki Boutabaa
  • Affiliation: Laboratoire de Mathématiques Pures, Université Blaise Pascal, (Clermont-Ferrand), Les Cézeaux, 63177 Aubiere Cedex, France
  • Email: boutabaa@ucfma.univ-bpclermont.fr
  • Alain Escassut
  • Affiliation: Laboratoire de Mathématiques Pures, Université Blaise Pascal, (Clermont-Ferrand), Les Cézeaux, 63177 Aubiere Cedex, France
  • MR Author ID: 64090
  • Email: escassut@ucfma.univ-bpclermont.fr
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 2557-2568
  • MSC (1991): Primary 11Q25
  • DOI: https://doi.org/10.1090/S0002-9939-98-04533-X
  • MathSciNet review: 1468183