# NMAK19009U Solving Polynomial Equations

MSc Programme in Mathematics

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.

**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.

Academic qualifications equivalent to a BSc degree is recommended.

Exercise sessions combine theoretical exercises with practical exercises using mathematical software.

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- Credit
- 7,5 ECTS
- Type of assessment
- Written assignment, 27 hours27-hour take home exam. It partly requires solving exercises with mathematical software.
- Exam registration requirements
A mandatory assignment must be approved before the exam. The assignment will be due in week 5 of the course.

- Aid
- All aids allowed
- Marking scale
- 7-point grading scale
- Censorship form
- No external censorship
One internal examiner
- Re-exam
Same as the ordinary exam.

To be eligible for the re-exam, students whose mandatory assignment have not been approved must re-submit the assignment. The mandatory assignment 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.

- Category
- Hours
- Lectures
- 21
- Theory exercises
- 28
- Preparation
- 130
- Exam
- 27
- Total
- 206

### Course information

- Language
- English
- Course code
- NMAK19009U
- Credit
- 7,5 ECTS
- Level
- Full Degree Master
- Duration
- 1 block
- Placement
- Block 1
- Schedule
- A
- Course capacity
- No restrictions/ no limitation
- Continuing and further education
- Study board
- Study Board of Mathematics and Computer Science

##### Contracting department

- Department of Mathematical Sciences

##### Contracting faculty

- Faculty of Science

##### Course Coordinators

- Elisenda Feliu (6-6768676e6b77426f63766a306d7730666d)
- Beatriz Pascual Escudero (7-65686477756c7d437064776b316e7831676e)