Abstract
Let \(\mathbb{K}\) be a complete algebraically closed p-adic field of characteristic zero. We give a new Nevanlinna-type theorem that lets us obtain results of uniqueness for two meromrphic functions inside a disk, sharing 4 bounded functions CM. Let P be a polynomial of uniqueness for meromorphic functions in \(\mathbb{K}\) or in an open disk, let f, g be two transcendental meromorphic functions in the whole field \(\mathbb{K}\) or meromorphic functions in an open disk of \(\mathbb{K}\) that are not quotients of bounded analytic functions and let α be a small meromorphic function with respect to f and g. We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove a result of uniqueness for functions: we show that if f′P′(f) and g′P′(g) share α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P′ satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k + 2 or n ≥ k + 3 used in previous papers by a new Hypothesis (G). Another consists of using the new Nevanlinna-type Theorem.
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Based on talk at the International Workshop on p-Adic Methods for Modelling of Complex Systems, Bielefeld, April 15–19, 2013.
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Escassut, A., Boussaf, K. & Ojeda, J. New results on applications of Nevanlinna methods to value sharing problems. P-Adic Num Ultrametr Anal Appl 5, 278–301 (2013). https://doi.org/10.1134/S2070046613040031
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DOI: https://doi.org/10.1134/S2070046613040031