Solving systems of polynomial equations

Although polynomial equations are much more complicated than linear equations, they still have a lot of structure, and there are many algorithms to bound the number of solutions and to describe their solution sets. Due to their simple structure, polynomials are often used in mathematical modelling, and this is just one reason why they abound in applications. The lecture will present different ideas from algebra and analysis to study and solve polynomial equations.

Some topics I plan to cover are:

• Polynomials in a single variable: How to count the roots (over C and over R), and how to find them.
• Polynomial systems with finitely many solutions; that is, zero-dimensional polynomial ideals.
• Systems of sparse polynomials (fewnomials).
• Primary decomposition of polynomial ideals.
• Real polynomial equations and sums of squares.

The main reference for the course will be Bernd Sturmfels' book that carries the same name as the lecture.

Date and time info
Thursday 11.00 - 12.30

Keywords
polynomials, Gröbner bases, fewnomials, commutative algebra

Prerequisites
Knowledge of elementary commutative algebra will be helpful, but not essential

Audience
MSc students, PhD students, Postdocs

Language
English