**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