《计算微分代数引论》课程主页
主讲老师： 李伟 副研究员
课程简介:
本课程为符号计算相关专业的博士研究生的专业课。微分代数是用代数方法研究微分方程解的结构的一门学科，是计算代数几何在代数微分方程领域的推广。本课程主要选讲常微分方程的代数理论与算法。通过本课程的学习，希望学生能够了解微分代数的一些基础知识与构造性算法，包括微分簇与微分理想的对应关系，微分Hilbert零点定理，微分簇的不可约分解，初等函数的积分问题等。本课程授课以自写讲义为主，参考书目如下：
J.F. Ritt, Differential Algebra, Amer. Math. Soc., New York, 1950
I. Kaplansky, An Introduction to Differential Algebra, Paris, Hermann, 1957.
E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, NewYork-London, 1973.
课程讲义*：
- Basic notions of differential algebra
- Differential rings and differential ideals[讲义 1]
- Decomposition of radical differential ideals[讲义 2]
- Differential Polynomial Rings and Differential Varieties
- Differential Characterisitic Sets[讲义 3]
- Ritt-Raudenbush Basis Theorem[讲义 4]
- Differential Algebra-Geomety Dictionary
- Ideal-Variety Correspondence in Differential Algebra [讲义 5]
- Differential Hilbert Nullstellensatz [讲义 6]
- Irreducible Decomposition of Differential Varieties[讲义 7]
- Extensions of Differential Fields
- Differential Primitive Theorem[讲义 8]
- Differential Transcendence Degree[讲义 9]
- Applications to Differential Varieties[讲义 10]
- Symbolic-integration for Elementary Functions
- Elementary Function and Order Function[讲义 11]
- Symbolic Integration of Elementary Functions[讲义 12]
- Algorithms and Problems in differential Elimination Theory
- Wu-Ritt Well-ordering Principle for differential polynomials [讲义 13]
- Differential decomposition Algorithms[讲义 14]
* A complete version of lecture notes written by Lei Fu via latex can be found here[讲义]