Computer Algebra

Class overview

This lecture course will provide Master students in Math with some basic concepts of Gröbner bases and applications.

Organization:

+ 3 hours/ week (including exercise sessions)

+ Time of Lecture Course: 14:00-16:30 

+ Exercise Class: 8:00-10:30

+ Room: Building L, Third Floor, Seminar Room 2 (Department of Math)

+ Written final examination (90-120 minutes)

Literature

[1] M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer - Verlag, 2008.

[2] D.A. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, 2015. 

[5] A. Bandini, P. Gianni, E. Sbarra, Commutative Algebra through Exercises, Springer, 2024 

Contents of the lecture course

Chapter I: Polynomial Rings

1.  The Univariate Polynomial Ring

2.  Multivariate Polynomial Rings

3.  Monoideals and Monomial Ideals

4.  Term Orderings

5.  Leading Term Ideals

6.  The Division Algorithm

Chapter II: Gröbner Bases

7.  Gröbner bases

8.  Polynomial Reductions

9.  Gröbner bases and Syzygies

10.  Buchberger’s Algorithm and Reduced Gröbner Bases

Chapter III: Applications of Gröbner Bases

11.   Elementary Operations on Ideals

12.  Elimination and Implicitization

13.  Hilbert’s Nullstellensatz

14.  Systems of Polynomial Equations

15.  Graph Colorings and Integer Programming (optional)

 Course Materials

• Lecture 1 and Exercises

• Lecture 2 and Exercises

• Lecture 3 and Exercises

• Lecture 4 and Exercises

• Lecture 5 and Exercises

• Lecture 6 and Exercises

• Lecture 7 and Exercises

• Lecture 8 and Exercises

• Lecture 9 and Exercises

• Lecture 10 and Exercises

Some Further Notes