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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.03548 |
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| _version_ | 1866914302890344448 |
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| author | Li, Jiyou Zhang, Zhiyao |
| author_facet | Li, Jiyou Zhang, Zhiyao |
| contents | Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$.
Denote the image set of this polynomial as $V_{f}=\{f(α)\midα\in\mathbb{F}_{q}\}$ and denote the cardinality of this set as $N_{f}$. A much sharper bound for $N_{f}$ is established in this paper. In particular, for any $p\neq 2, 3$, and for nearly every generic quartic polynomial $f \in \mathbb{F}_{q}[x]$, we obtain $$\lvert N_f - \frac{5}{8} q \rvert \leq \frac{1}{2}\sqrt{q} + \frac{15}{4},$$ which holds as a simple corollary of the main result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03548 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Improving bounds for value sets of polynomials over finite fields Li, Jiyou Zhang, Zhiyao Number Theory 11T06, 14H05 Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $f \in \mathbb{F}_{q}[x]$ be a polynomial of degree $d > 0$. Denote the image set of this polynomial as $V_{f}=\{f(α)\midα\in\mathbb{F}_{q}\}$ and denote the cardinality of this set as $N_{f}$. A much sharper bound for $N_{f}$ is established in this paper. In particular, for any $p\neq 2, 3$, and for nearly every generic quartic polynomial $f \in \mathbb{F}_{q}[x]$, we obtain $$\lvert N_f - \frac{5}{8} q \rvert \leq \frac{1}{2}\sqrt{q} + \frac{15}{4},$$ which holds as a simple corollary of the main result. |
| title | Improving bounds for value sets of polynomials over finite fields |
| topic | Number Theory 11T06, 14H05 |
| url | https://arxiv.org/abs/2601.03548 |