(Created 2009-08-11.)

COMPUTER ALGEBRA | FMA115 |

**Aim**

The aim of the course is to introduce some basic concepts and algorithms on which modern computer algebra is based, and to explain how computer algebra programs, e.g. Maple, work and how they can be applied effectively. The course should also give basic knowledge in classical algebra. The aim is also to help the student develop his och her ability in problem solving.

*Knowledge and understanding*

For a passing grade the student must

be able to explain how some central, and in applications important, computer algebra algorithms work.

show good knowledge about those basic concepts in abstract algebra which are required in order to understand and use the algorithms treated in the course.

*Skills and abilities*

For a passing grade the student must

in the context of problem solving using computer algebra programs be able to explain how the basic algorithm works with the specific problem.

in writing as well as orally show good ability independently to explain mathematical reasoning in a structured and logically clear way.

**Contents**

- Main algorithms: representation and efficient handling of numbers, polynomials and rational functions. Factorization in Z, Z[x] and Zp[x].
- Gröbner bases and nonlinear systems of equations. Practical applications.
- Isolation of real roots. Sturm sequences, continued fractions.
- Modular methods: Hensel lifting, Berlekamp's algorithm.
- Symbolic summation and integration. Gosper's algorithm.

**Literature**

Childs, L.N.: A Concrete Introduction to Higher Algebra, Springer 2000. ISBN 0-387-98999-4.

Lecture notes with supplementary material.