Introduction to Algebraic Geometry and Applications
Code
10836
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
João Pedro Bizarro Cabral
Total hours
22
Teaching language
Português
Objectives
Introduce the students to the basics of Algebraic Geometry and its applications in fields like integer programming, multivariate polynomial splines and algebraic coding theory.
Prerequisites
Linear Algebra I and II, Geometry, Mathematical Analyis III and IV.
Subject matter
1. Review of the basic concepts of polynomials, rings, ideals and quotient ideal.
2. Affine varieties and Gröbner bases: Monomial orders, polynomial division, Gröbner bases, affine varieties, elimination theory.
3. Resolution of systems of polynomial equations using elimination theory.
4. Integer programming: Integer programming problems, how to solve these using Gröbner bases .
5. Multivariate polynomial splines: Modules over rings, Gröbner bases for modules over rings, geometry of polytopes, polyhedral complex, applications of Gröbner bases of modules over rings to multivariate polynomial splines theory.
6. Algebraic Coding Theory: Finite dimensional algebras, finite fields, error correcting codes, cyclic codes, Reed-Solomon decoding algorithms.
Bibliography
1. D.Cox& al., ‘’Using Algebraic Geometry’’, Grad. TextsinMath. (Springer)
2. D. Cox, J.Little&D.O’Shea, “Ideals, varieties and algorithms”, (Springer)
3. D. Perrin, ‘’Algebraic Geometry, an introduction ‘’, Universitext (Springer)
4. Greuel G., Pfister G., “ A Singular Introduction to Commutative Algebra”,(Springer)
Teaching method
Tutorial orientation of the student, with indication of which matters to study, problems to solve resolution and clarification of doubts about theory and problems.
Evaluation method
Students must deliver all works assigned for evaluation.
Final grade will be given by an average of the grade given to each work.