《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。
课程旨在介绍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,Wu-Ritt特征列方法与微分簇不可约分解算法等。

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, New York-London, 1973.

    时间地点:

    每周二6-8节(14:20-17:00);中关村教学楼N306教室。

    课程讲义:

    1. Basic notions of differential algebra

      • Differential rings and differential ideals[讲义 1]
      • Decomposition of radical differential ideals[讲义 2]

    2. Differential Polynomial Rings and Differential Varieties

    3. Differential Algebra-Geomety Dictionary

      • Ideal-Variety Correspondence in Differential Algebra
      • Differential Hilbert Nullstellensatz [讲义 5]
      • Irreducible Decomposition of Differential Varieties[讲义 6]

    4. Extensions of Differential Fields

    5. Algorithms and Constructive Methods for Algebraic Ordinary Differential Equations

      • Wu-Ritt Well-ordering Principle for differential polynomials [讲义 11]
      • Differential decomposition Algorithms[讲义 12]
      • Applications to Mechanical Theorem Proving and Discovering[讲义 13]

    6. Decompostion Algorithms for Algebraic Partial Differential Equations[讲义 14]