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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An, T.T.H., Escassut, A. Meromorphic solutions of equations over non-Archimedean fields. Ramanujan J 15, 415–433 (2008). https://doi.org/10.1007/s11139-007-9086-9
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DOI: https://doi.org/10.1007/s11139-007-9086-9