The Problem Of Value Distribution In P-Adic Field

In1920’s, R. Nevanlinna, the mathematician of Finland, created the Nevanlinna theory in complex number field. Since then, scholars all over the world got many of beautiful results with the application of Nevanlinna the-ory. This days, with the research of number theory in p-adic field, scholars tried to study the generalization of Nevanlinna theory in p-adic field. The analogue of first and second main theorem in p-adic field has been researched by Ha H.K.[12], Ha H.K.&My V.Q.[13], A. Boutabba[14] and C. Corrales-Rodriganez[15]. In2000, P.C. Hu and C.C. Yang gave a comprehensive intro-duction about the value distribution theory in their book Meromorphic Func-tions over Non-Archimedean Field, which showed the maturity of the research about this theory.In this thesis, I show some work about the problem of value distribution in p-adic field, which I studied under the guidance of my supervisor professor P.C. Hu. The main contents are as follow:In Chapter1, we describe some basic results of Nevanlinna Theory in p-adic field and some other concepts and results.In Chapter2, we consider the Hayman’s conjecture in p-adic field. We study some work of J. Ojeda[2], and we obtain Theorem1. Let f∈M(k) be transcendental and deg(A)≥deg(B). Let k, m be integer, and m> k+1. If and it exists a sequence of zeros of f which has order not less than k, and the absolute value of these zeros be tend to infinity. Then f(k)+Tfm has infinitely many zeros that are not zeros of f.Theorem2. Let f∈M(k) be transcendental and deg(A)≥deg(B)(resp. Let f∈Mu(d(0,R-))). Let k, m be integer. If m≥k+4, and the order of all zeros of f be not less than k, then f(k)+Tfm has infinitely many zeros that are not zeros of f.In Chapter3, we consider the problem of exceptional values of p-adic functions and derivatives. We study some work of A. Escassut and J. Ojeda[3], and we obtainTheorem3. Let f∈M(κ) be not constant, have two pole at least and have no pole of order≥2. Then f" have no exceptional value.