Introduction to Computer Algebra

The lecture gives an introduction to basic concepts of Computer algebra. In more detail, we will discuss the following topics:

• Integer and polynomial arithmetics, fast multiplication, gcd computations, complexity of algorithms.
• Systems of equations, Hermite- and Smith-normal form, p-adic approximation, lattices, LLL algorithm.
• Resultants and extended gcd computations.
• Multivariate polynomials, Gröbner bases for ideals and modules, modular methods, Faugère's F4 and F5 algorithm, syzygies, free resolutions.
• (Squarefree) Factorization, Hensel lifting, factorization with LLL, primality tests, factorization over algebraic number fields.

References

• Cohen: A Course in Computational Algebraic Number Theory
• Cox, Little, O'Shea: Ideals, Varieties, and Algorithms
• von zur Gathen: Modern Computer Algebra
• Greuel, Pfister: A SINGULAR introduction to Commutative Algebra

Date and time info
Wednesday 13:15 - 14:45, Thursday 9:15 - 10:45

Keywords
Arithmetics, Systems of (non-)linear equations, lattices, resultants, Gröbner bases, factorization

Prerequisites
Basic knowledge about algebra and the usage of computer algebra systems

Audience
MSc students, PhD students, Postdocs

Language
English