On Nevanlinna's Four-value Theorem And Some Related Uniqueness Problems

In the 1920's. the mathematician Rolf Nevanlinna, a Finlander. founded one of the most important theories of the twentieth century, the value distribution theory of meromorphic functions over the open complex plane C, which is usually called Nevanlinna theory in honor of him. This theory is composed of two main theorems, which are called Nevanlinna's first and second main theorems that had been significant breakthroughs in the development of the classical function theory, since the latter generalizes and extends Picard's first theorem greatly, and hence it denoted the beginning of the theory of meromorphic functions. For some eighty years. Nevanlinna theory has been well developed in itself and been widely applied to the researches of the unicity of meromorphic functions, normal families, complex dynamics and differential equations etc..Meanwhile, the modern high dimensional complex analysis has progressed at a, fast speed, which is mainly represented by the theories of complex geometry and complex manifolds, and followed by the rapid growth of the theory of meromorphic maps over the m-dimensional complex vector space C^m and certain complex manifolds. On the other hand, the functional analysis over p-adic. i.e., non-Archimedean, fields has also developed quickly during the last half a, century. In view of the beauty and exactitude of Nevanlinna theory. many outstanding mathematicians in the fields of complex geometry, complex manifolds, algebraic geometry and number theory etc. such as L. Ahlfors, H. Cartan, H. Weyl, J. Weyl, S.S. Chern, W. Stoll. H. Wu. Y.T. Siu, P. Griffiths, J. King, J. Carlson. M. Cowen, A. Vitter, B. Shiffman, H. Fujumoto, J. Noguchi, S. Lang, P. Vojta, P.M. Wong. M. Ru, W. Cherry, Z. Ye and K. Yamanoi etc. founded and developed consecutively the value distribution theories of holomorphic curves and meromorphic maps over certain complex manifolds and p-adic fields to protective varieties, respectively. As special cases to those beautiful and marvellous theories, we could derive the corresponding ones for meromorphic functions over C^mand p-adic fields, respectively.In 1929. Rolf Nevanlinna applied his value distribution theory to consider the conditions under which a meromorphic function of a single variable could be determined, and derived the famous Nevanlinna's five-value, four-value and three-value theorems. Initialed by these three results, the unicity of meromorphic functions of a. single variable has been drastically studied by lots of mathematicians during the past eighty years and gradually consummated in itself, which was recently extended to those of meromorphic functions of several variables and over p-adic fields, respectively.The present thesis involves those results of the author that investigate the unicity of meromorphic functions over C, C^m andκ(any algebraically closed p-adic field of characteristic zero, complete for some non-trivial non-Archimedean absolute value). respectively, under the guidance of Professor Pei-Chu Hu and Professor Hong-Xun Yi. It consists of four parts and the matters are explained as below.Chapter 1 studies the unicity of meromorphic functions over C sharing three pairwise distinct values, and a fourth value or a pair of distinct values with truncation, which improves Nevanlinna's four-value theorem that has been continuously worked on by G.G. Gundersen, E. Mues, H. Ueda and G. Brosch etc.. Also, at the end of this part, we derive a result associated to Nevanlinna's three-value theorem.Chapter 2 continues the study of the previous one and concerns the unicity of an entire function over C, its first derivative and its linear differential polynomial of rational coefficients under the condition of sharing a non-zero polynomial, which takes advantage of Wiman-Valiron's estimate and generalizes many known results.Albeit Chapters 1 and 2 are deeply related to one another and might be read in succession, Chapter 3 handles a relatively independent problem, that is, the unicity of meromorphic functions over C^m concerning unique range sets. Our results show that there exists one kind of triple-unique-range-sets of six pairwise distinct elements totally, which, together with an example to gauge the lower bound, reduces the current estimate c₃(M(C^m))≤9 by P.C. Hu - C.C. Yang to 5≤_c3(M(C^m))≤6.Chapter 4 deals with the unicity of a p-adic meromorphic function f overκand its generated linear differential polynomial P[f] of general type under the condition of sharing two distinct finite values inκsystematically, and several sufficient conditions for the identity f≡P[f] are provided. Finally, we also talk about the existence of global meromorphic solutions to p-adic linear differential equations overκ.