NMAK16003U Computational Algebraic Geometry

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. The study of the geometric objects defined by the solutions to polynomial equations is called algebraic geometry.

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: Gröbner bases, elimination theory, resultants, techniques for finding and classifying the roots of polynomials in one variable, implicit and parametric descriptions of varieties, finite dimensional algebra and zero dimensional ideals, and, if time permits, also homotopy methods for numerically solving polynomial equations will be discussed.

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 preparation for the master course "Algebraic Geometry", as it gives a practical and hands-on approach to the topic.