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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
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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

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

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