The Polynomial Method and extensions
In 2006 Dvir came up with an extremely elegant way to prove that any "Kakeya" set over finite fields, i.e., a set that contains a line in every direction, must be very large. Specifically if we consider a Kakeya set in F_q^n - the n-dimensional space over a field of size q - then the Kakeya set needs to be of size at least roughly q^n/(n!). The proof, surprisingly, uses properties of low-degree multivariate polynomials!
In this talk I will describe the method and some of our improvements which use properties of low-degree polynomials and their derivatives.
Based on joint work with Zeev Dvir, Swastik Kopparty, and Shubhangi Saraf.