Centre for Mathematical Sciences

Lund University

 

Advanced intensive mini-course

 

Exploring algebraic sets with computer algebra system


Nacer Makhlouf (
Mulhouse, France)

Times and Place: MH 228, Matematikum, Lund University

Monday 27/3,                     kl 15-17

Wednesday 29/3,               kl 13-15

Thursday 30/3,                   kl 10-12

 

Language of lectures: English

Contact person: Sergei Silvestrov, MH:154, (+46) 46  222 4010,

sergei.silvestrov@maths.lth.se    http://www.maths.lth.se/matematiklth/personal/ssilvest/index.html

 

Summary:

The power of computers and the discovery of new algorithms for dealing with polynomial equations have sparked a minor revolution in the study and practice of algebraic geometry.

It has given rise to some new applications of algebraic geometry mainly in robotics.

Polynomials appear in almost all areas in scientific computing and engineering. Most of the applications need to solve equations involving polynomials and system of polynomials, often in many variables. The major fields where polynomial systems are used are robotics, Computer Aided design and modelling, signal processing and filter design. The wide range of use of polynomial systems needs to have fast and reliable methods to solve them. There is two general approaches: symbolic and numeric. The symbolic approach is based on  algebraic geometry and the theory of Groebner basis.

Algebraic geometry is the study of geometries that come from algebra, in particular, from ring. In classical algebraic geometry, the algebra is the ring of polynomials and the geometry is the set of zeros of polynomial, called an algebraic variety. According to the Hilbert basis theorem, a finite number of polynomial suffices. The geometry we are interested in concerns affine varieties, which are curves, surfaces and higher dimensional objects defined by polynomial equations.

The study of affine algebraic varieties is carried to the study of the ideals of polynomial ring of several variables over fields. The most commonly used fields are: the rational numbers for computer examples, the real numbers for drawing pictures of curves and surfaces and the complex numbers for proving many theorems.

The goals of the lectures are to provide an introduction to algebraic geometry by investigating

the link between algebra and geometry and to highlight the applications of Groebner basis.

Plan

1. Exploration of affine algebraic varieties with Mathematica system
and basic properties

2. Ideal and ideal -variety correspondence (Hilbert's
Nullstellensatz theorem, irreducible varieties...)

3.
Groebner basis and applications

4.
Elimination and extension  theories