Meromorphic solutions of equations over non-Archimedean fields |
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Authors: | Ta Thi Hoai An Alain Escassut |
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Affiliation: | (1) Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, 10307 Hanoi, Viet Nam;(2) Laboratoire de Mathématiques, UMR 6620, Université Blaise Pascal (Clermont-Ferrand), Les Cézeaux, 63177 Aubiere-Cedex, France |
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Abstract: | In this paper, we give some conditions to assure that the equation P(X)=Q(Y) has no meromorphic solutions in all K, where P and Q are polynomials over an algebraically closed field K of characteristic zero, complete with respect to a non-Archimedean valuation. In particular, if P and Q satisfy the hypothesis (F) introduced by H. Fujimoto, a necessary and sufficient condition is obtained when deg P=deg Q. The results are presented in terms of parametrization of a projective curve by three entire functions. In this way we also obtain similar results for unbounded analytic functions inside an open disk. |
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Keywords: | Nevanlinna theory Functional equations Uniqueness polynomials Meromorphic functions Curve Singularity |
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