涉及差分的亚纯函数唯一性的若干研究及其应用

The uniqueness theory of meromorphic functions is an important part of the Nevanlinna value distribution theory. The uniqueness problem related to derivatives is one of the important research directions, and has attracted the domestic and foreign experts and scholars' attention. The uniqueness problem related to differences is a new hot-spot in recent years. It is a natural product when Nevanlinna value distribution theory was extended to the difference field, and has large research space and unique research value.The project team will draw on classical research methods of uniqueness of meromorphic functions , combing with the difference equations theory and the theory of normal family, to carry out the following researches:.(1) investigations on the Brück conjecture related to differences ;.(2) investigations on the uniqueness problems related to small functions and sets in the difference field, and applications to complex difference equations.

亚纯函数唯一性理论是Nevanlinna值分布理论的重要组成部分。涉及导数的唯一性问题是其中一个重要研究方向，一直受到国内外专家学者的关注。涉及差分的唯一性问题是近几年来出现的新的研究热点，是Nevanlinna值分布理论被推广到差分领域并深入发展的自然产物，具有较大的研究空间和独特的研究价值。本项目组将借鉴经典的亚纯函数唯一性研究方法，并结合差分方程理论和正规族理论，开展以下研究工作：.(1) 研究涉及差分的Brück猜想；.(2) 研究差分中涉及小函数或集合的唯一性问题，并应用于研究复差分方程。

本项目主要开展了以下工作：(1) 研究了亚纯函数与其差分多项式分担小函数或集合的唯一性问题，并取得了一定的进展；(2) 研究了差分方程，得到了两类非线性差分方程亚纯解的性质，推广了Malmquist关于非线性微分方程的相关结论。