In the winter term 2018/19 I will read a lecture course

Differential Algebra

(with exercises) for bachelor and master students in mathematics. Prerequisite is only knowledge from a basic course in algebra, as one can obtain in the lecture course Grundlagen der Algebra und Computeralgebra. Knowledge about Gröbner bases are of advantage but not necessary.

The course will mainly introduce the Ritt-Kolchin theory of differential polynomials. Thus the main emphasis will be on differential ideal theory, which extends the classical algebraic theory of ideals in commutative rings to differential polynomials, and related algorithms. In the last years, the Ritt-Kolchin theory could help in the solution of many algorithmic problems based on sets of algebraic differential equations. It is of interest for both pure and applied mathematics. If time permits, we will also discuss some basic notions of the differential Galois theory of linear ordinary differential equations.

Literature:

• Hubert: Notes on Triangular Sets and Triangulation-Decomposition Algorithms. I. Polynomial Systems. II. Differential Systems, in Winkler, Langer (eds) Symbolic and Numerical Scientific Computation, Springer
• Kolchin: Differential Algebra and Algebraic Groups, Academic Press
• Magid: Lectures on Differential Galois Theory, AMS
• Ritt: Differential Equations from the Algebraic Standpoint, AMS
• Ritt: Differential Algebra, Dover
• Sauloy: Differential Galois Theory Through Riemann-Hilbert Correspondence, AMS
• Singer, van der Put: Galois Theory of Linear Differential Equations, Springer
• Sit: The Ritt-Kolchin Theory for Differential Polynomials, in: Guo, Keigher, Cassedy, Sit (eds) Differential Algebra and Related Topics, World Scientific

Course Dates:

Time/Place: Tuesday 11:00-13:00, SR 0450a, Wednesday 09:00-11:00, SR 2420
Start: Oktober 23rd

Exercises

I will read the exercises for the course myself. They are every Wednesday 11:00-13:00 in SR 0450. The exercise sheets will be available here. The first sheet will be published on October 30th after we have actually started with differential algebra.

• 1. exercise sheet (to be discussed on 7 November)
• 2. exercise sheet (to be discussed on 21 November)
• 3. exercise sheet (to be discussed on 28 November)
• 4. exercise sheet (to be discussed on 5 December)
• 5. exercise sheet (to be discussed on 12 December)
• 6. exercise sheet (to be discussed on 16 January)
• 7. exercise sheet (to be discussed on 23 January)
• 8. exercise sheet (to be discussed on 30 January)
• Printout of the Maple session in the last lecture