涉及分担函数的正规族与某类复差分多项式值分布的研究

Normal family theory of Meromorphic functions is the basic content of the study of complex dynamical systems. Since its establishment, it is a research concern in the field of one variable complex function. Currently, there exist a lot of research results. This project is mainly using the classical theories and methods of complex analysis to research a number of issues in normal family theory and complex difference theory. The project will be the depth and expansion of these directions of research work, On the basis of preliminary results, we further study some of their public issues, especially the criterion of normal family of meromorphic functions which sharing the same function. In recent years, the value distribution theory of complex difference is one of the hot points which concerned by complex analysis and physics scholars, the value distribution of some special complex difference polynomials is very popular subjects in this fields. We hope progress in these issues can attract domestic and foreign scholars' attention, and we believe our topic has good development prospects.

亚纯函数正规族理论是研究复动力系统的基础内容,自建立以来就成为单复变函数领域中备受关注的研究方向，目前已经有大量的研究成果。本项目主要运用复分析中的经典理论和方去法去研究正规族理论以及复差分理论中的一些问题。本项目将深入和拓展这两个方向某些问题的研究工作，在前期结果的基础上进一步研究一些问题，特别是亚纯函数情况下的亚纯函数族分担同一个函数的正规定则。最近几年，复差分值分布理论是受到复分析和物理两方面学者关注的热点之一，一些特殊的复差分多项式值分布问题是当今比较热门的课题。我们希望这些问题的进展能吸引国内外学者关注，并且相信我们的课题是有良好的发展前景的。

本项目给出了如下几个方面的结果: 1.研究涉及分担值、分担集合、或者分担函数的亚纯函数族正规族相关问题，取得了一些成果，改进了现有的结果。2.在对某类特殊复差分多项式值分布方面的研究，也完成论文1篇，已投稿在审。3.对亚纯函数分担函数唯一性方面的研究也取得了一些成果。4.在对复化数理方程精确解方面的研究，我们对Fisher方程，Boiti-Leon-Pempinelli （BLP）方程组等方面的研究发现了一些新解。目前，已发表论文5篇，还有投稿在审1篇，已录用待发表2篇，完成了本项目的任务。