tailieunhanh - Nevanlinna five-value theorem for p-adic meromorphic functions and their derivatives

Nevanlinna five value theorem for p adic meromorphic functions and their derivatives. In this paper, we gave a result similar to the Nevanlinna five-value theorem. | Nguyễn Xuân Lai Tạp chí KHOA HỌC & CÔNG NGHỆ 81(05): 91 - 95 NEVANLINNA FIVE-VALUE THEOREM FOR P-ADIC MEROMORPHIC FUNCTIONS AND THEIR DERIVATIVES Nguyen Xuan Lai Hai Duong College ABSTRACT In this paper, we gave a result similar to the Nevanlinna five-value theorem. Keywords: Unique problem, p-adic Meromorphic functions, derivative, Nevanlinna, Height of p-adic meromorphic functions INTRODUCTION* In 1920, Nevanlinna proved the following result (the Navanlinna four- value theorem): Theorem A. Let f and g be two non-constant meromorphic functions. If f and g share four distinct values CM, then f is a Mobius transformation of g. In 1997 Yang ang Hua [17] studied the unicity problem for meromorphic functions and differential monomials of the form HEIGHT OF P-ADIC MEROMORPHIC FUNCTIONS Let f be a nonzero holomorphic function on £ p . For every a Î £ p , expanding f as f = ∑ Pi ( z − a ) polynomials Pi of define For a point d Î f f′ ≠ 0. be a non- zero finite value. If g n g′ some (n+ 1)- th root of unity d, or and g = c2e − cz c1 , c2 f = c1ecz for three non-zero constants and c such that ( c1c2 ) n +1 c 2 = −a 2 . In this paper, by using some arguments in [10], [16] and the Nevanlinna theory in onedimensional non-archimedean case, developed in [6], [12], [13], we gave a result similar to the Nevanlinna five-value theorem. * degree i around a, we p, we define function p → by v df (a ) = v f −d (a ) Fix real number r with 0 k 0 v ( z) = v f ( z ) if v f ( z ) ≤ k ≤k f and n ≤f k (r ) = ∑ v ≤f k ( z ) , For a subset S of n ≤f k (a, r ) = n ≤f k−a (r ) Fix real number r with 0 k f N ( a, r ) , p, define m f (∞, r ) = max {0,log f £ Let a is an element of m f (a, r ) = m p, r }. define 1 ( f −a ) (∞,r ) , N f (a, r ) = N f1 −af2 (r ) , and N f (∞, r ) = N f2 (r ) , H f (r ) = max H fi (r ) . In a like manner we define N ( a, r ) , we set Define z ≤r N ( a, r ) , p and If a = 0, then set , kf (a, r ) . N l , f