涉及复微分差分和Tropical的值分布与函数方程研究

Based on the theory of meromorphic functions and tropical meromorphic functions, using the methods on value distribution and its applications, we will consider four problems as follows: the zeros and uniqueness of complex differential-difference polynomials; the properties of solutions on Fermat type complex differential-difference equations; tropical Nevanlinna theory and its applications on ultra-discrete equations; non-linear tropical Nevanlinna theory. Our research purpose is to construct differential-difference analogues or tropical analogues of some classical problems and to explore the similarities or differences on research methods and conclusions in different fields. This project is devoted to consider the value distribution and functional equations on the combiniation of complex differential with complex difference and the tropical meromorphic functions and its application on ultra discrete equations and to explore the innovation on nonlinear tropical Nevanlinna theory. These investigations can bring new vitality on value distribution and functional equations theories. Early, the applicant and members had good research works in complex differential, complex difference and complex differential-difference fields. The applicant also has obtained some preliminary results on tropical meromorphic functions theory. We believe that we can get some significant results in this project. Thus, we can enhance and raise our level of research in the field of complex differential-difference and develop the corresponding theory on tropical meromorphic functions in China.

本项目以亚纯函数和Tropical（热带）亚纯函数理论为基础，运用值分布理论及其应用的研究方法，拟开展四个方面的问题研究：复微分差分多项式零点与唯一性、费马型复微分差分方程解的性质、热带Nevanlinna理论及其在超离散方程的应用、非线性热带Nevanlinna理论；研究目标是通过构建若干经典问题的复微分差分和热带版本，探索不同领域间研究方法和结果的统一性和差异性。本项目是融合复微分和复差分进行值分布和函数方程的研究，是率先在国内开展热带亚纯函数理论及在超离散方程应用的研究，并在非线性热带问题进行创新，给值分布和复方程理论注入了新的研究活力。项目团队前期在复微分、复差分、复微分差分的值分布与函数方程等相关问题有较好研究基础，申请人在热带亚纯函数领域取得了初步研究成果，相信我们通过本项目可获得一些重要的创新成果，能提升项目团队在复微分差分领域研究实力，促进热带亚纯函数相关理论在国内的发展。

团队成员按照申请书预设的研究内容在复微分差分的值分布与函数方程方面取得了丰富的研究成果，从成对型和指数型多项式的角度得到了复微分差分的Hayman猜想的结果，拓展了对于函数增长级条件的限制；得到了亚纯函数导数和差分分担公共值的若干结果；研究了比如推广型Fermat差分方程、一阶非线性差分方程、几类线性或者非线性复微分差分方程的亚纯函数解的性质，我们的研究成果也得到了众多后续研究人员的关注，引领了复微分差分领域的研究。在Tropical理论的研究中，目前还处于攻关和撰写完善成果的阶段。总体来讲，项目能够对预设的研究内容进行较好的研究和攻关，取得了丰富的研究成果。依托本项目发表了16篇相关的学术论文，并获得与本项目有关的后续国家自然科学基金项目1项，江西省自然科学基金1项，邀请了6位复分析专家学术报告，参加了2次中芬复分析会议，依托本项目，培养5名硕士毕业生，目前还有4名在读硕士研究生。