Cartan理论和复域函数方程解的研究

The functional equations have important applications in the fields such as physics, medicine, etc. Cartan’s second main theorem as a generalization of Nevanlinna theory is a strong result in the value distribution of holomorphic curves in the higher dimensional complex projective space, as well as an efficient tool for certain problems specially problems on some functional equations in the complex plan. As a result, we intend to study the following problems in this project by applying the existing theories of Nevanlinna-Cartan, Wiman-Valiron, convex hull and complex function spaces, etc: 1. the generalizations of Cartan's second main theorem for Wronskian determinant and its difference analogue for Casorati determinant; 2. the relationships between solutions and coefficients of linear differential equations of certain types in the complex plane and the unit disc; 3. the solutions of nonlinear difference equations of certain types such as properties of exponential polynomial solutions, value distribution of meromorphic solutions, difference analogue of Super-Fermat problem; 4. the classifications of some nonlinear q-difference equations which contain Painlevé type. This project will enhance the intersections of different directions of mathematics, develop and rich researches on both Cartan theory and complex functional equations, give the theoretical basis for the fields such as physics, and has important scientific significance.

复域函数方程在物理学、医学等众多领域有广泛应用。Cartan第二基本定理作为Nevanlinna理论的推广是高维复射影空间中全纯曲线值分布的重要结果，同样也为复平面上特定问题特别是某些函数方程问题的解决提供了有效工具。鉴于此，本项目将利用现有Nevanlinna-Cartan理论、Wiman-Valiron理论、凸包理论和复函数空间理论等拟主要研究以下内容：1.Cartan第二基本定理Wronskian行列式和其差分模拟Casorati行列式的推广；2.复平面和单位圆盘上几类线性微分方程解与系数的关系；3.几类非线性差分方程的解，如指数多项式解特点、亚纯解值分布、差分Super-Fermat问题；4.含Painlevé型非线性q-差分方程的分类。本项目的实施将加强不同数学分支之间的交叉，丰富发展Cartan理论和复域函数方程的研究，为物理学等领域提供理论基础，有重要科学意义。

复域函数方程在物理学、医学等众多领域有着广泛应用。Cartan第二基本定理作为Nevanlinna理论的推广是高维复射影空间中全纯曲线值分布的重要结果，同样也为复平面上特定问题特别是某些函数方程问题的解决提供了有效工具。鉴于此，在本项目中，我们应用现有Nevanlinna-Cartan、Wiman-Valiron等理论重点研究了以下内容：(1)研究了复域内两类非线性微分方程的超越亚纯解；(2)研究了复域内两类(Fermat型和Super-Fermat型)差分方程的整函数解；(3)研究了复域内一类非线性微分-差分方程的超越整函数解。本项目所研究的问题都是目前国内外热点问题，通过该项目的研究，获得若干比较有意义的成果，其中一些研究工作具有重要的理论和应用价值。

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