In a paper by Ellenberg and Gijswijt on the cap set problem, the proof of proposition 2 relies on the claim that certain matrices have rank 1. This is not obvious to me. Why?
https://arxiv.org/pdf/1605.09223v1.pdf
The matrices in question are indexed by a subset $A\subset \mathbb{F}_q^n$, with $a,b$ entry
$$m(a)F(b)$$
where $m$ is a monomial of degree at most $d\leq 2n$, and $F$ a polynomial of degree at most $2n-d$. Furthermore, each variable appears with degree at most $q-1$ in any term of $m(x)F(x)$