Polynomial Corners Over finite Fields

Abstract

Recently there has been some progress in understanding the density of a subset of $[N{]}^2$ that avoids polynomial patterns. Kravitz, Kuca, and Leng showed that if $P\in \mathbb{Z}[z]$ satisfies certain conditions, then any set $A\subseteq [N{]}^2$ does not contain $(x,y),(x+P(z),y),(x,y+P(z))$, we must have \[ |A|\ll_P\frac{N^2}{(\log\log\log N)^c} \] for some small constant $c$. In this article, we show a similar result in $({\mathbb{F}}_p{)}^2$ where we get a better bound on the density of a set $A\subseteq ({\mathbb{F}}_p{)}^2$ not containing $(x,y),(x+P(z),y),(x,y+P(z))$ with some conditions on $P\in {\mathbb{F}}_p[z]$.