微分多项式分解的算法和理论研究

Differential is a subject developed in 1950s mainly by J.F.Ritt and his student. In the view of algebra, tools and skills from algebraic geometry and symbolic computation are used to deal with differential equations of polynomial form. As the main object in differential algebra, the decomposition of differential polynomials has become an important topic in the area. It is closely related to some classical mathematical problems and as subproblems, decomposition of algebraic polynomials and factorization of differential operators has been widely explored. In this project, we will focus on the properties, practical and more efficient algorithms and theory of decomposition of univariate ordinary differential polynomials and the related Lüroth's generators problem of mediate fields. Also, we will study the problem of differential polynomial generators of simple transcendental extensions of differential fields and try to generalize the uniqueness theorems for maximal decompositions of algebraic and Ore polynomials to the differential case. The results of the project will become some elementary part of differential algebra and have applications in simplifying equations, factorization of Ore polynomials, proper parameterization of curves and surfaces, differential Galois theory and some other issues in differential algebra.

微分代数是上世纪五十年代由J.F.Ritt等人建立并发展起来的学科，它站在代数的观点，利用代数几何和符号计算中的工具和技巧来处理多项式形式的微分方程。作为微分代数中主要研究对象的微分多项式，其分解问题则是该学科中的重要研究课题，并与一些经典的数学问题密切相关；而其特殊而重要的子类- - 代数多项式和微分算子的分解，也已被广泛研究。本项目将致力于研究单变元常微分多项式分解的性质、算法和极大分解的唯一性理论以及与之密切相关的中间域Lüroth生成元等问题。给出切实可行、复杂度更好的算法，探讨与代数多项式和Ore多项式体系分解理论中极大分解"唯一性"定理相对应的结论，并利用分解研究和计算微分域的单超越扩张和中间域的微分多项式生成元。该项目的结果将成为微分代数中重要理论结果的一部分，并对有关方程的化简求解，Ore多项式的分解，曲线与曲面的正则参数化，微分Galois理论等问题有较大的影响和帮助。