Complex and p-Adic Meromorphic Functions f'P'(f), g'P'(g) Sharing a Small Function

Abstract

Let K be a complete algebraically closed p-adic field of characteristic zero. We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f P (f) and g P (g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k + 2 or n ≥ k + 3 used in previous papers by Hypothesis (G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with (q − 1) times the characteristic function of the considered meromorphic function. Notation and definition: Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value |. |. We will denote by E a field that is either K or C. Throughout the paper we denote by a a point in K. Given R ∈]0, +∞[ we define disks d(a, R) = {x ∈ K | |x − a| ≤ R} and disks d(a, R −) = {x ∈ K | |x − a| < R}. A polynomial Q(X) ∈ E[X] is called a polynomial of uniqueness for a family of functions F defined in a subset of E if Q(f) = Q(g) implies f = g. The definition of polynomials of uniqueness was introduced in [19] by P. Li and C. C. Yang and was studied in many papers [11], [13], [20] for complex functions and in [1], [2], [9], [10], [17], [18], for p-adic functions. Throughout the paper we will denote by P (X) a polynomial in E[X] such that P (X) is of the form b l i=1 (X − a i) ki with l ≥ 2 and k 1 ≥ 2. The polynomial P will be said to satisfy Hypothesis (G) if P (a i) + P (a j) = 0 ∀i = j. We will improve the main theorems obtained in [5] and [6] with the help of a new hypothesis denoted by Hypothesis (G) and by thorougly examining the situation with p-adic and complex analytic and meromorphic functions in order to avoid a lot of exclusions. Moreover, we will prove a new theorem completing the 2nd Main Theorem for p-adic meromorphic functions. Thanks to this new theorem we will give more precisions in results on value-sharing problems. 0 2000 Mathematics Subject Classification: 12J25; 30D35; 30G06. 0