亚纯函数差分多项式值分布问题的研究

The project intends to use Nevanlinna theory of meromorphic functions to investigate value distribution problems related to difference polynomial of meromorphic functions. Based on difference counterparts of Nevanlinna theory, we will seek some new crossing points among value distribution theory，complex differential equations and complex difference equations. Furthermore, we will try to look for a general way to solve the problems in difference counterparts of Nevanlinna theory and get some results to rich continuous development of the difference theory in recent years.In this project, we mainly continue to study the related proplems of value distribution by some innovative research methods, which can be positive in the developments and connections in different branch of mathematics. The study of this project is the cross-over study between Nevanlinna theory and complex difference equations theory, and is an interesting subject of complex analysis recently.

本项目拟利用Nevanlinna理论研究亚纯函数差分多项式的值分布问题。申请者将在Nevanlinna理论的基础上，寻求其与差分方程理论的交叉点，希望通过对亚纯函数值分布理论、复微分方程，复差分方程等理论的深入研究，找到解决相关差分方程问题的一般方法和途径，争取得到几个比较理想的结果，丰富近几年来不断发展的差分Nevanlinna理论的内涵，同时创新研究方法，可以进一步深入研究相关问题。本项目是Nevanlinna 理论和复差分理论的交叉研究，是近几年来复分析方向比较活跃的研究领域，对复分析的发展、探索和促进不同数学分支间的交叉都很有意义。

本项目利用Nevanlinna理论对一类亚纯函数差分方程的解的存在性和增长性问题进行了研究。我们证明了该类亚纯函数方程存在超级小于1的亚纯解，解决了计划书中的问题，并且给出了解的增长性结论。本项目的研究成果丰富了复微分差分Nevanlinna理论的研究成果，改进了一些已有结果，同时对于复分析和不同学科交叉研究具有重要的科学意义。项目研究成果已整理完成，正在投稿中。