Course Syllabus for

# Algebraic Geometry and Solving Systems of Polynomial Equations Algebraisk geometri och lösning av system av polynomekvationer

## FMA150F, 7.5 credits

Valid from: Spring 2018
Decided by: Professor Thomas Johansson
Date of establishment: 2018-08-24

## General Information

Division: Mathematics
Course type: Third-cycle course
Teaching languages: English, Swedish

## Aim

The aim of the course is to prepare postgraduate students for research using Gröbner bases for solving and interpreting systems of polynomial equations in several variables mainly within algebraic geometry.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must be able to explain the concept of a Gröbner basis and describe why they are useful for solving systems of polynomial equations.

Competences and Skills

For a passing grade the doctoral student must

• be able to reproduce key results and give rigorous and detailed proofs of them,
• be able to compare key results,
• be able to apply the basic techniques, results and concepts of the course to concrete examples and exercises,
• be able to combine concepts from the course with other important topics in algebra.
• be able to use different methods for solving and for interpreting systems of polynomial equations.

## Course Contents

Affine varieties and ideals in the ring of polynomials. Gröbner bases. Elimination theory. Algebraic-Geometric Correspondences. Polynomial and Rational Functions on a Variety.

## Course Literature

• Cox, D. & Little, J.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 2016. ISBN 9783319374277.
• Cox, D. & Little, J.: Using Algebraic Geometry. Springer Science & Business Media, 2005. ISBN 9780387207339.

The books are available as e-books via the Mathematics library.

## Instruction Details

Types of instruction: Lectures, exercises, project. If the number of participants is small, the course is given as a reading course.

## Examination Details

Examination formats: Written exam, oral exam, written assignments. Take-home exam.