微分代数簇有理通解的相关理论及算法

Algebraic variety is the main research object in algebraic geometry. It is more general than curves and surfaces. The prametrization of curves and surfaces has attracted extensive attention of researchers for a very long time. Many approaches have been proposed for solving this problem. Much of differential algebra or differential algebraic geometry can be regarded as a generalization of the algebraic geometry theory to the analogous theory for the differential equations. Recently, there has been growing interest concern about extending or applying the related research results in algebraic geometry into the differential case, e.g., parametrization of differential varieties. Furthermore, research on rational general solutions of differential equations or systems is the basis of the parametrization of differential varieties. The aim of this proposal is to study some relevant theories and algorithms of rational general solutions of nonlinear differential equations or systems based on the above mentioned curve or surface parametrization methods in algebraic geometry. It mainly contains the following three aspects: properties of rational (general) solutions of first order multivariables autonomous differential systems; necessary and sufficient conditions for a higher order algebraic differential equation having a rational general solution; and application of computing rational solutions to solving some linear or nonlinear differential equations or systems.

代数簇是代数几何的中心研究对象，曲线与曲面，作为最简单的代数簇，其参数化问题长期以来一直受到计算几何学界的广泛关注，并已取得大量的研究成果。微分代数几何理论可看作是代数几何中相关理论的推广，因此延伸和应用代数几何中的成熟结果到微分的情形，成为近年来众多学者研究的热点。目前，微分代数几何中微分代数簇的参数化较受关注，该问题的研究引发了微分方程或系统有理通解的计算。本项目主要借助以曲线曲面参数化为工具的代数几何方法，研究与计算非线性微分方程或系统有理通解相关的一些理论及算法，主要包括以下三方面工作：一阶多变元自治微分系统有理(通)解的性质；高阶代数微分方程存在有理通解的充分必要条件；有理解的计算在求解某些线性或非线性微分方程或系统中的应用。

曲线与曲面，作为代数几何中最简单的代数簇，其参数化问题长期以来一直受到计算几何学界的广泛关注，并已取得大量的研究成果。微分代数几何理论可看作是代数几何中相关理论的推广，因此延伸和应用代数几何中的成熟结果到微分的情形，成为近年来众多学者研究的热点。目前，微分代数几何中微分代数簇的参数化较受关注，该问题的研究引发了微分方程或系统有理通解的计算。本项目主要研究了与计算非线性微分方程或系统有理通解相关的一些理论与算法，主要包括以下成果：给定有理通解次数界的情形下，一阶多变元自治微分系统存在有理通解的充分必要条件；一阶自治微分方程的线性可解性以及对应到相同不变代数空间曲线的不同有理解之间的关系；一阶多变元自治线性多项式微分系统有理解的表示及其有理通解的次数界；二阶微分方程关于有理可解性的分类；基于三角分解的最优控制切换面的计算。项目组共发表论文 8 篇，其中 SCI 期刊论文两篇，EI 期刊论文三篇，中文核心期刊论文三篇。