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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References
T. T. H. An, “Defect relation for non-Archimedean analytic maps into arbitrary projective varieties,” Proc. Amer.Math. Soc. 135, 1255–1261 (2007).
K. Boussaf, “Motzkin factorization in algebras of analytic elements,” Ann. Math. Blaise Pascal 2 (1), 73–91 (1995).
K. Boussaf, A. Escassut and J. Ojeda, “Complex and p-adic branched functions and growth of entire functions,” Bull. BelgianMath. Soc. Simon Stevin 22, 781–796 (2015).
A. Boutabaa, “Théorie de Nevanlinna p-adique,” Manuscripta Math. 67, 251–269 (1990).
A. Boutabaa and A. Escassut, “URS and URSIMS for p-adic meromorphic functions inside a disk,” Proc. EdinburghMath. Society 44, 485–504 (2001).
K. S. Charak, “Value distribution theory of meromorphic functions,” Math. Newsletter 18 (4), 1–35 (2009).
J. F. Chen, “Uniqueness of meromorphic functions sharing two finite sets,” paper to appear.
A. Escassut, Value Distribution in p-Adic Analysis (WSCP Singapore, 2015).
A. Escassut and T. T. H. An, “p-AdicNevanlinna theory outside of a hole,” VietnamJ. Math. 45 (4), 681–694 (2017).
H. K. Ha, “On p-adic meromorphic functions,” Duke Math. J. 50, 695–711 (1983).
M. O. Hanyak and A. A. Kondratyuk, “Meromorphic functions in m-punctured complex planes,” Matematychni Studii 27 (1), 53–69 (2007).
Vu Hoai An, Pham Ngoc Hoa and Ha Huy Khoai, “Value sharing problems for differential polynomials of meromorphic functions in a non-Archimedean field,” paper to appear.
M. Krasner, “Prolongement analytique uniforme et multiforme dans les corps valués complets,” Les tendances géométriques en algèbre et théorie des nombres, Clermont-Ferrand, pp.94–141 (1964). Centre National de la Recherche Scientifique (1966). (Colloques internationaux de C. N.R. S. Paris, 143).
Liu Gang and Meng Chao, “Uniqueness for the difference monomials of p-adic entire punctions,” paper to appear.
E. Motzkin, “La décomposition d’un élément analytique en facteurs singuliers,” Ann. Inst. Fourier 27 (1), 67–82 (1977).
X. G. Qi, L. Z. Yang and K. Liu, “Uniqueness and periodicity of meromorphic functions concerning the difference operator,” Comput. Math. Appl. 60, 1739–1746 (2010).
K. Yamanoi, “The secondmain theorem for small functions and related problems,” ActaMath. 192, 225–294 (2004).
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Escassut, A., An, T.T.H. New Applications of the p-Adic Nevanlinna Theory. P-Adic Num Ultrametr Anal Appl 10, 12–31 (2018). https://doi.org/10.1134/S2070046618010028
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DOI: https://doi.org/10.1134/S2070046618010028