# Proceedings of the American Mathematical Society

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## On uniqueness of $p$-adic meromorphic functionsHTML articles powered by AMS MathViewer

by Abdelbaki Boutabaa and Alain Escassut
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.
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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