NMAK19009U Solving Polynomial Equations

Volume 2020/2021
Education

MSc Programme in Mathematics

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 present techniques to study the structure and properties of the zero set of a system of polynomial equations (an algebraic variety), as well as introduce algorithms and methods to solve systems of polynomial equations. The main topics of the course are: 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 homotopy methods for numerically solving polynomial equations. The theory will be applied to real-life systems.

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.

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 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.

Algebra 2

Academic qualifications equivalent to a BSc degree in mathematics is recommended.
3 hours of lectures and 4 hours of exercises for 7 weeks.
Exercise sessions combine theoretical exercises with practical exercises using mathematical software.
  • Category
  • Hours
  • Lectures
  • 21
  • Preparation
  • 84
  • Theory exercises
  • 28
  • Exam
  • 73
  • Total
  • 206
Continuous feedback during the course of the semester
Credit
7,5 ECTS
Type of assessment
Written assignment, 3 days
Oral examination, 20 minutes
Written take-home assignment in week 8 of the course + oral examination (without preparation) on the written assignment in week 9 of the course
Exam registration requirements

Two mandatory assignments must be approved before the exam. The assignments will be due in weeks 3 and 5 of the course.

Aid
All aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
Two internal examiner
Re-exam

27-hour take-home assignment.

To be eligible for the re-exam, students whose mandatory assignments have not been approved must re-submit the assignment. The mandatory assignments must be approved no later than 3 weeks before the re-exam week.

Criteria for exam assesment

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