**Computer Algebra (326.031)**

**Winter semester 2022/2023**

**Carsten Schneider**

Time: Thurdays, 10:15 - 11:45

Room: HS 12

In the last decades big parts of mathematics has been algorithmized and many mathematical problems (or problems coming from natural and technical sciences that can be modeled in mathematics) can be solved with the computer. A major contribution for this algorithmic revolution is the computer algebra. This lecture aims at introducing the most crucial algorithms in this field and illustrating how they can be used for non-trivial applications.

We discuss constructive symbolic methods for simplification of expressions and solving algebraic (i.e., polynomial) systems of equations. Among others, the following algorithms are explored:

- basic structures and algorithms
- the extended Euclidean algorithm and applications (Chinese Remainder Theorem, Pade approximation, Rational function and number reconstructions, partical fraction decomposition, BCH codes)
- modular versions of the (extended) Euclid algorithm and properties of resultants
- a gentle introduction to Gröbner bases
- maybe: symbolic summation and integration
Some slides accomplishing the lecture can be found in the pdf-file CA.pdf. In addition, some concrete examples discussed in the lexture can be also found within the Mathematica notebook Examples.nb

The lecture is partially based on

- Modern Computer Algebra by Joachim von zur Gathen and Jürgen Gerhard
- Algorithms for Computer Algebra by Keith O. Geddes, Stephen R. Czapor, George Labahn
- Polynomial Algorithms in Computer Algebra by Franz Winkler
- Ideals, Varieties, and Algorithms by David A. Cox, John Little, Donal O’Shea

**Exercises:**

Time: Thurdays, 12:00 - 12:45

Room: HS 12

Exercise instructor: Günter Landsmann (landsmann@risc.jku.at)

The exercises start on October 13. Always the week before homeworks are posed on an exercise sheet that will be accessible below. At the beginning of the exercise class the participants indicate which homeworks they have carried out. Based on that the participants are selected in order to present the solutions. The homeworks themselves consist of concrete calculations by using a CA system, proving some simple statements that are needed in the lecture, or considering further aspects that are not treated in the lecture. In particular, little programming exercises might be posed.

List of exercise sheets

for 13.10.2022: ca-ex-01.pdf for 20.10.2022: ca-ex-02.pdf for 27.10.2022: ca-ex-03.pdf for 03.11.2022: ca-ex-04.pdf for 10.11.2022: ca-ex-05.pdf for 17.11.2022: ca-ex-06.pdf for 24.11.2022: ca-ex-07.pdf for 01.12.2022: ca-ex-08.pdf for 15.12.2022: ca-ex-09.pdf for 12.01.2023: ca-ex-10.pdf for 19.01.2023: ca-ex-11.pdf for 26.01.2023: ca-ex-12.pdf