Algebraic Geometry Tools for Polynomial Systems in Engineering

Aim: Algebraic geometry studies the solution sets of polynomial equations using techniques from algebra and geometry. In this mini-course of 4 sessions we will develop a basic toolkit of algebraic geometry concepts and methods useful for studying polynomial systems in engineering. Each two-hour meeting is part lecture and part exercises. Participants will meet basic building blocks such as varieties, irreducibility, dimension, fibers of maps and a glimpse at intersection theory. A goal is to introduce and understand recent results by Leonie Kayser and Sibren Lagauw on counting the number of solutions of optimization problems arising from systems & control.

Content: The sessions will address the following themes:

• Affine and projective varieties, Hilbert’s Nullstellensatz
• Dimension of varieties, fibers, and intersections
• Degree of varieties and Hilbert functions, Bézout’s Theorem
• Varieties of low-rank matrices and counting solutions via the Thom—Porteous formula

Target Audience: This course is intended for researchers and students who want to understand and work with polynomial equations arising in applications. No prior background in algebraic geometry or engineering is required. Familiarity with linear algebra and basic properties of multivariate polynomials is sufficient.

Suggested further reading: The following texts may be enjoyable as an elementary-ish introduction to algebraic geometry

• David A. Cox, John Little, and Donal O’Shea, Ideals, Varieties, and Algorithms
• Emily Clader and Dustin Ross, Beginning in Algebraic Geometry
• Andrew J. Sommese and Charles W. Wampler, The Numerical Solution of Systems of Polynomials
• Pinaki Mondal, How Many Zeroes?

Schedule