Groebner basis and Buchberger algorithm.
Elimination of variables.
Introduction to algebraic varieties.
Projective varieties.
The dimension and its computation.
D.Cox, J.Little, D.O'Shea, Ideals, Varieties and Algorithms, Springer 1992, capp. 1,2,3,4,8,9
On Moodle page are available the course notes.
Learning Objectives
Constructive introduction to Algebraic Geometry. Learn methods and computational techniques to manage polynomials and polynomial systems. Computer Algebra. Appreciate the differences between the linear and the nonlinear case.
Prerequisites
Linear Algebra and Analytic Geometry. Polynomials, groups, Rings and Ideals. Projective Space.
Topological Spaces.
Teaching Methods
Lectures. Discussion of examples and exercises. The course has parallel computer sessions with software Macaulay2 (M2)
http://www.math.uiuc.edu/Macaulay2/
Type of Assessment
Oral exam. Intermediate tests at Computer Science Laboratory with M2 sessions. Alternatively, at the oral exam will be requested the presentation of some M2 exercises, listed at the end of the course.
Course program
Groebner basis and Buchberger algorithm. Noetherian rings and BasisSatz. Monomial orders and division algorithm. Monomial ideals and Groebner basis. Buchberger criterion and Buchberger algorithm. Membership problem in ideals.
Elimination of variables.
Elimination Theore. Intersection of ideals, lcm and GCD. Radical Ideal.
Introduction to algebraic varieties. Algebraic Varieties, the correspondence Ideals-Varieties. Zariski Topology. Nullstellensatz. The resultant. Extension Theorem. Geometric interpretation of elimination and Closure Theorem.
Graph colorability.
Parametrization of algebraic varieties.