New Applications of the p-Adic Nevanlinna Theory

  • Alain EscassutEmail author
  • Ta Thi Hoai An
Research Articles


Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fn(x)fm(ax + b), gn(x)gm(ax + b) with |a| = 1 and nm, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |nm| ≥ 5, then fn(x)fm(ax + b) − w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fn(x)fm(ax + b).

Key words

p-adic meromorphic functions Nevanlinna’s theory values distribution small functions Picard values branched values 


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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Université Clermont AuvergneLaboratoire de Mathématiques Blaise PascalAUBIEREFrance
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  3. 3.Institute of Mathematics and Applied Sciences (TIMAS)Thang Long UniversityHanoiVietnam

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