Expanding Polynomials and Combinatorial Geometry
by Dr Frank de Zeeuw
Abstract: The polynomial f=x+y has the special property that on certain sets its image is relatively small; for instance, if A is an arithmetic progression, then f(AxA) has size 2|A|-1. Similarly, g=xy has small image on a geometric progression. It is natural to ask which other polynomials have this property, and which polynomials instead "expand", in the sense that their image is always significantly larger than the input.
In this talk, I will survey what is known about this question, which goes back to Erdős, Szemerédi, and Elekes, and was developed by Bourgain, Tao, and others. Although the question seems purely algebraic (with a combinatorial flavor), the partial answers that we have tend to rely on geometry. Specifically, expanding polynomials are closely related to a subfield of combinatorial geometry known as incidence geometry. This topic has recently seen great progress due to algebraic methods introduced by Guth, Katz, and Rudnev, leading to many new results on expanding polynomials. I will discuss these developments, including some contributions of my own.