Ranges of polynomials control degree ranks of Green and Tao over finite prime fields

Abstract

Let $p$ be a prime, let $1\le t<d<p$ be integers, and let $S$ be a non-empty subset of ${\mathbb{F}}_p$. We establish that if a polynomial $P:{\mathbb{F}}_p^n\to {\mathbb{F}}_p$ with degree $d$ is such that the image $P(S^n)$ does not contain the full image $A({\mathbb{F}}_p)$ of any non-constant polynomial $A:{\mathbb{F}}_p\to {\mathbb{F}}_p$ with degree at most $t$, then $P$ coincides on $S^n$ with a polynomial that in particular has bounded degree-$\lfloor d/(t+1)\rfloor$-rank in the sense of Green and Tao. Similarly, we prove that if the assumption holds even for $t=d$, then $P$ coincides on $S^n$ with a polynomial determined by a bounded number of coordinates.

25 pages

References in corpus (1)

• Partition and Analytic Rank are Equivalent over Large Fields