Solving Polynomial Equations

Course content

Multivariate polynomial equations are omnipresent in real-life applications. For instance, they appear in models in chemistry, biology, economy and robotics. Solving polynomial equations is often a difficult task and leads to interesting geometric, algebraic and algorithmic questions.

In this course we will introduce current algorithmic and practical methods to solve polynomial equations and to study the main class of geometric objects: algebraic varieties. We will discuss among others: Gröbner bases, elimination theory, resultants, techniques for finding and classifying the roots of polynomials in one variable, implicit and parametric descriptions of varieties, zero-dimensional solution sets, Newton polytope, semi-algebraic sets, and, if time permits, homotopy methods for numerically solving polynomial equations. The theory will be applied to real-life systems in the exercise sessions.

The students' mastering of this field will serve as a good background for both further theoretical studies within algebraic geometry, and also for practical real-life applications outside academia. In particular, this course can serve as a good complement for other master courses in Algebraic Geometry, as it gives a practical and hands-on approach to the topic.

Education

MSc Programme in Mathematics

Learning outcome

Knowledge: The students are able to define, describe the main properties of, and use in practical situations the following: algebraic varieties, Gröbner bases, elimination theory, resultants, techniques for finding and classifying the roots of polynomials in one variable, implicit and parametric descriptions of varieties, zero-dimensional systems, Newton polytope, semi-algebraic sets, and eventually homotopy methods in numerical algebraic geometry.

Skills: By the end of the course the students are able to use and implement methods to find and describe solutions to polynomial equations using available mathematical software. The students are able to understand the difference between the methods, their theoretical foundations, what they are best suited for, identify their limitations, and choose the appropriate method in each situation.

Competences: By the end of the course the students will have developed a theoretical and practical understanding of the main aspects and current trends in the field of solving polynomial equations, and be able to use this knowledge in theoretical contexts and in applications.

3 hours of lectures and 4 hours of exercises for 7 weeks.
Exercise sessions combine theoretical exercises with practical exercises using mathematical software.

Algebra 2

Academic qualifications equivalent to a BSc degree is recommended.

Continuous feedback during the course of the semester
ECTS
7,5 ECTS
Type of assessment
Written assignment, 27 hours
27-hour take home exam. It partly requires solving exercises with mathematical software.
Aid
All aids allowed
Marking scale
Censorship form
No external censorship
One internal examiner
Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 21
• Theory exercises
• 28
• Preparation
• 130
• Exam
• 27
• English
• 206