HUJI Combinatorics Seminar
When: Monday January 10th, 2022, at 4PM (Israel time)
Zoom link: https://huji.zoom.us/j/88356751646?pwd=RGN1ZUI2eTBlREM0SmNjMEtCYW9EQT09
Speaker: Lisa Sauermann (MIT)
Title: Finding distinct-variable solutions to linear equations in F_p^n
Fix a prime p and system of linear equations wth coefficients in F_p. For large n, what is the largest size of a subset A of F_p^n which does not contain a solution to this system in which the variables are distinct vectors in A? A famous special instance of this question is the problem of bounding the largest size of a subset of F_p^n without a three-term arithmetic progression. In 2016, Ellenberg and Gijswijt made a breakthrough on this problem, and shortly afterwards Tao introduced the so-called slice rank polynomial method as a reformulation (and generalization) of their argument. Unfortunately, for other instances of the question above, the slice rank polynomial method cannot handle the distinctness condition for the variables. In this talk, we will discuss two different results concerning certain instances of this question. 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.