Ranges of polynomials control degree ranks of Green and Tao over finite prime fields
Abstract
Abstract Let $p$ be a prime, let $1 \le t \lt d \lt p$ 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.
Fields of Study & Subjects
Coding theory and cryptography
Finite Group Theory Research
Analytic Number Theory Research
Publication details
- Journal / source
- Proceedings of the Edinburgh Mathematical Society
- Publisher
- Cambridge University Press (CUP)
- Pages
- 1–28
- Publication date
- 18 February 2026
- Language
- EN