Ideals, Varieties and Algorithms

Content

Objective

The purpose of this course is to learn about Groebner basis which is useful to solve some problems on algebraic varieties, especially for the solution of systems of equations, to understand how to use it in the proof of theorem extension.

Course Content

Ring theory and fields (summary), Polynomial rings and affine space, Affine varieties, Parametrization, Ideals, One variable polynomials;
Monomial orders, Division algorithm, Monomial ideals and Dickson's lemma, Hilbert bases theorem, Groebner bases, Properties of Groebner bases, Buchbergers algorithm, Applicaitons of Groebner bases;
Elimination and Extension theorems, Resultants and the extension theorem.

Course Learning Outcomes

To know how to calculate a Groebner base for an ideal by using the algorithm of Buchberger
To know how to use the Groebner bases to solve the appearence problem for an ideal
To be able to use the Groebner bases in the theory of elimination to solve the system of equations

Teaching and Learning Methods

Lecture, discussion, problem solving

References

Ideals, Varieties and Algorithms, D. Cox, J. Little, D. O’Shea.

Assessment System

ECTS