复微分差分方程问题及Fermat型方程亚纯解的值分布

This project is based on the hot research topic in value distribution of meromorphic function. Using the uniqueness theory, the complex differential equation, the complex difference equation, the Wiman Valiron theory, and partial differential equation as research tools, we mainly studies the following problems: (1) Estimate the existence of meromorphic solutions of complex differential and difference equations. Study the forms and properties of the meromorphic solutions of the equations. Include the Painlevé、Ricatti type of complex equations. Meanwhile, study the the problem of value distribution of the meromorphic solutions of those equations; (2) The relations of characteristic functions of meromorphic functions and their differences, the uniqueness problem of meromorphic functions sharing values or small functions with their differences, such as the difference analogy of the Brück conjecture and the problem of Yi-Yang ; (3) Estimate the existence of meromorphic solutions of one or two complex variables of Fermat type and the partial differential equation. Consider the uniqueness problem of the meromorphic solutions sharing values or sets with meromorphic functions. This project is the expansion of value distribution theory of meromorphic functions. It includes the crossover study among the uniqueness problem of meromorphic functions，complex functions and the partial differential equation. In addition, it can generate many novel research questions and has many good prospects and enough space for further research. By this project , we expect to obtain some important research results.

本项目立足于亚纯函数值分布论中的热门前沿问题。我们将以亚纯函数唯一性、复微分方程、复差分方程、Wiman-Valiron理论、偏微分方程理论等为工具，主要研究以下问题：(1)复微分、差分方程亚纯解的存在性、解的形式和性质等，包括差分Painlevé、Ricatti方程，并研究方程亚纯解的值分布问题；(2)亚纯函数与其差分特征函数的关系，并讨论它们分担值、小函数的唯一性问题, 如Brück猜想和Yi-Yang问题的差分模拟等；(3)一维和二维Fermat型方程和一类复偏微分方程亚纯解的存在性问题，亚纯解与其它亚纯函数分担值、集合的唯一性问题。本项目是亚纯函数值分布论的深化和拓展，并包含亚纯函数唯一性与复方程问题的交叉研究，唯一性理论与偏微分方程理论的交叉研究，讨论的问题新颖，具有很好的前景。我们有望通过本项目获得一些重要的研究结果。

本项目立足于亚纯函数值分布论中的热门前沿问题。我们以亚纯函数唯一性、复微分方程、复差分方程、Wiman-Valiron理论、偏微分方程理论等为工具，主要研究、解决或者部分解决了以下问题：(1)复微分、差分方程亚纯解的存在性、解的形式和性质等，包括差分Painlevé、Ricatti方程，并研究方程亚纯解的值分布问题；(2)亚纯函数与其差分特征函数的关系，并讨论它们分担值、小函数的唯一性问题， 如Brück猜想的差分模拟；(3)一维和二维Fermat型方程和一类复偏微分方程亚纯解的存在性问题和拟素问题；(4)Gamma 函数与Serberg 类中的L-函数的代数微分无关性问题。本项目深化和拓展了亚纯函数值分布论，交叉研究了亚纯函数唯一性与复方程问题，唯一性与偏微分方程问题。共计发表相关科研论文15篇，SCI收录14篇，中文核心1篇。