复微分差分多项式的值分布与复微分差分方程

Based on some classical problems in value distribution (the zeros and uniqueness of meromorphic functions) and some important functional equations (Fermat equations,linear equations), we will construct some appropriate differential-difference analogue theory of above. Using Nevanlinna theory as a research tool, through the studying of the zeros and uniqueness of differential-difference polynomials, the existence of meromorphic solutions of complex differential-difference equations, the relationship of the growth of solutions with the equations coefficients,the convergence of zeros and poles of meromorphic solutions, we will explore the similarities and differences among the complex differential, complex difference with complex differential-difference by similar problems and equations. Even though the reseach contents of this project are closely linked with the single field, but we will confront more difficulties when dealing with the problems in differential-difference. We need to explore new research methods and tools. So it is worth paying a lot of effort to investigate. This project is a research in combiniation with complex differential and complex difference. In addition, it can generate many novel research questions and has many good prospects and enough space for further research. Early, the applicant had good research works in complex differential and complex difference fields and had a depth discussion on the value distribution of difference polynomials and difference equations. In this project, we expect to get some important findings in the field of complex differential-difference.

本项目以值分布理论的一些经典问题(亚纯函数的零点问题、唯一性问题)和重要的函数方程(Fermat方程、线性方程)为载体，拟构建相应的复微分差分的模拟理论。我们将利用Nevanlinna理论作为主要工具，通过研究复微分差分多项式的零点与唯一性、复微分差分方程亚纯解的存在性、解的增长性与方程系数的关系、亚纯解的零点及极点收敛指数等内容，探索同类问题或方程在复微分、复差分、复微分差分领域间的共性与差别。项目的研究内容与单个领域的研究关系密切，但往往问题处理更具有复杂性，需要探索新的研究方法，值得努力攻关。此项目是复微分、复差分的彼此融合，研究的问题新颖，具有很好的研究前景和拓展空间。前期申请人在复微分和复差分两个领域都有较好的研究基础，并在复差分多项式的值分布和复差分方程理论方面进行过深入探讨，我们有望通过本项目获得一些复微分差分领域重要的研究结果。

本项目按照研究计划，开展了复微分差分多项式以及复微分q-差分多项式值分布的研究，建立了目前此问题的最佳条件和结果；开展了几类具有代表性类型的复微分差分方程的亚纯函数解的性质研究，得到了方程亚纯函数解在存在性、增长性方面的一些结果；开展了函数和它的微分差分混合算子、函数导数与其差分分担公共值或者公共小函数的问题研究，建立了此类问题唯一性的结果；开展了热带亚纯函数理论的研究，建立了经典的Fermat方程、Bruck猜想、Hayman猜想的热带版本理论。在微分差分领域结合的研究，后续有很多研究学者也开始关注此类问题，热带亚纯函数领域作为一个复分析的拓展也开始得到更多的关注，我们的成果具有很好的引领作用。本项目预定的研究计划顺利，并且后续挖掘了新的问题和研究方向进行了拓展研究，在项目的资助下获得后续国家资金资助3项，将对此项目进行后续的拓展研究。. 本项目的研究成果以论文的形式已发表11篇，其中SCI检索论文10篇，参加了4个国内外复分析的学术会议，资助1人赴东芬兰大学进行访问学习，邀请了5位国内外专家到我校学术交流；项目培养硕士研究生5名；指导南昌大学创新学分科研训练项目2项。