New Applications of the <Emphasis Type="Italic">p</Emphasis>-Adic Nevanlinna Theory |
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Authors: | Alain Escassut Ta Thi Hoai An |
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Institution: | 1.Université Clermont Auvergne,Laboratoire de Mathématiques Blaise Pascal,AUBIERE,France;2.Institute of Mathematics,Vietnam Academy of Science and Technology,Hanoi,Vietnam;3.Institute of Mathematics and Applied Sciences (TIMAS),Thang Long University,Hanoi,Vietnam |
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Abstract: | 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 n ≠ m, 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 |n ? m|∞ ≥ 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). |
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