Ranges of polynomials control degree ranks of Green and Tao over finite prime fields
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917292090064896 |
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| author | Karam, Thomas |
| author_facet | Karam, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11088 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ranges of polynomials control degree ranks of Green and Tao over finite prime fields Karam, Thomas Combinatorics Number Theory 11T06 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. |
| title | Ranges of polynomials control degree ranks of Green and Tao over finite prime fields |
| topic | Combinatorics Number Theory 11T06 |
| url | https://arxiv.org/abs/2305.11088 |