Masato Mimura（Tohoku University）「Tao's slice rank method for a system of equations. Part 1: Basics on the method, and main results.」
This is a joint work with Norihide Tokushige (Ryukyu).
Ellenberg and Gijswijt obtained an exponential bound for the maximal size of 3-AP-free subsets $A$ of $F_p^n$, which is based on a breakthrough by Croot–Lev–Pach. Tao reformulated the argument by defining the notion of 'slice ranks' of polynomials. In this first part, we overview how Tao's slice rank method works for this example. In this example, the subset $A$ is assumed to satisfy that the only solutions of the (single) equation $x-2y+z=0$ are singletons $(x,y,z)=(a,a,a)$, where $a$ is in $A$. We exhibit the precise statements of our main results, which treat a system of linear equations.
Part 2 (from 15:45 to 17:15, for those who have interest) : The proof of the main results.
We proceed to the proof of our main results. As a main example, we study the maximal size of a subset $A$ of $F_p^n$ that does not admit distinct five points $x,y,z,u,v$ in $A$ forming a 'W-shape', namely, $(x,y,z,u,v)$ is a solution of the system of two equations $x-y-z+u=0$ and $x-2z+v=0$. To deal with this problem, we adapt an argument of Sauermann to our case. In this second part, we describe the ideas of our argument for the case of a W-shape.