Nevanlinna-Cartan理论和复差分方程的研究

Difference Cartan’s second main theorem is a generalization of difference Nevanlinna second main theorem. It is a strong result in the value distribution of holomorphic curves in the higher dimensional complex projective space, and also provides an efficient tool for some problems in the complex plane. Complex difference equations have important applications in the fields such as medicine, physics, etc. As a result, we want to investigate the following things in this project by applying the existing Cartan-Nevanlinna theory: firstly, we intend to get further generalizations of difference Cartan’s second main theorem for the purpose of providing new tools in the field of complex analysis; secondly, we will further investigate the solutions of some difference equations with certain type, especially the difference painlevé III and painlevé V equations, and study the classification of non-linear q-difference equation which contains q-difference Painlevé V equations; last but not least important, by comparing the methodologies used in the study of meromorphic solutions of differential and (q-) difference equations, we try to find some tools as the good linear operator for the study of general classes of functional equations. This project will develop both Nevanlinna-Cartan theory itself and its applications in the complex difference equations, and is of great theoretical significance.

差分Cartan第二基本定理是差分Nevanlinna第二基本定理的推广，是高维复射影空间中全纯曲线值分布的重要结果，也为复平面上许多问题的解决提供了有效工具。复域差分方程在医学、物理学等众多领域应用广泛。鉴于此，本项目将应用现有的Nevanlinna-Cartan理论研究以下内容：(1). 探求差分Cartan第二基本定理的进一步推广，以便为复分析领域提供新的理论工具；(2). 研究某些特定类型差分方程解的性质，特别是 painlevéIII型和painlevéV型差分方程，研究包含painlevéV型非线性q-差分方程的分类；(3). 通过比较研究微分、(q-)差分方程亚纯解时所使用方法的异同，探求类似于好的线性算子的其他适用于研究一般函数方程解的有效工具。 本项目的实施将进一步深化Nevanlinna-Cartan理论及其应用，具有很高的理论意义。

本项目主要研究了Nevanlinna-Cartan理论及其在复域差分方程和亚纯函数唯一性理论中的应用。复域差分与差分方程在医学、物理学等众多领域应用广泛。本项目应用现有的Nevanlinna理论，得到了几种特定类型的差分方程亚纯解的性质；得到了超级小于一的整函数及其线性差分算子分担值问题的一些结果；关于Cartan理论的进一步推广和Nevanlinna-Cartan理论在其他型(q-)差分方程中应用的研究也取得了一些进展。本项目的实施进一步深化了Nevanlinna-Cartan理论及其应用，具有很好的理论意义。