The Ramanujan Journal

, Volume 15, Issue 3, pp 415–433 | Cite as

Meromorphic solutions of equations over non-Archimedean fields

  • Ta Thi Hoai An
  • Alain EscassutEmail author


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.


Nevanlinna theory Functional equations Uniqueness polynomials Meromorphic functions Curve Singularity 

Mathematics Subject Classification (2000)

12E05 11S80 30D35 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiViet Nam
  2. 2.Laboratoire de Mathématiques, UMR 6620Université Blaise Pascal (Clermont-Ferrand)Aubiere-CedexFrance

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