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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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