Finding distinct-variable solutions to linear equations in \mathbb{F}_p^n


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October 7, 2021 4:30 pm - 5:30 pm
Science Center Hall A
Address: 1 Oxford Street, Cambridge, MA 02138

Lisa Sauermann - MIT

For a fixed prime p and large n, what is the largest size of a subset of \mathbb{F}_p^n which does not contain a non-trivial solution to some given linear equation or system of linear equations? This is a fundamental question in additive combinatorics with a long history. While there are various different notions of "non-trivial solutions", in this talk we say that a solution is non-trivial if all of its variables are distinct.

A particularly famous and important instance of the question above is the problem of bounding the largest size of a subset of \mathbb{F}_p^n which does not contain a three-term arithmetic progression. In 2016, Ellenberg and Gijswijt made a breakthrough on this problem and the approach in their proof was later generalized by Tao yielding what is now called the slice rank polynomial method. Unfortunately, for other instances of the question above, the slice rank polynomial method cannot handle the condition for the variables in a non-trivial solution to be distinct.

In this talk, we will first give a brief survey on the slice rank polynomial method and some of its applications, and we will then discuss two different results concerning the question above. These results combine the slice rank polynomial method with additional combinatorial ideas in order to handle the distinctness condition. We also discuss an application of one of these results to the Erdös-Ginzburg-Ziv problem in discrete geometry.