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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.03732 |
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| _version_ | 1866910355429523456 |
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| author | Arala, Nuno Chow, Sam |
| author_facet | Arala, Nuno Chow, Sam |
| contents | We establish expansion properties for suitably generic polynomials of degree $d$ in $d+1$ variables over finite fields. In particular, we show that if $P\in\mathbb{F}_q[x_1,\ldots,x_{d+1}]$ is a polynomial of degree $d$ coming from an explicit, Zariski dense set, and $X_1,\ldots,X_{d+1}\subseteq\mathbb{F}_q$ are suitably large, then $|P(X_1,\ldots,X_{d+1})|=q-O(1)$. Our methods rely on a higher-degree extension of a result of Vinh on point--line incidences over a finite field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03732 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expansion properties of polynomials over finite fields Arala, Nuno Chow, Sam Combinatorics Number Theory 11B06, 11B30, 51B05 We establish expansion properties for suitably generic polynomials of degree $d$ in $d+1$ variables over finite fields. In particular, we show that if $P\in\mathbb{F}_q[x_1,\ldots,x_{d+1}]$ is a polynomial of degree $d$ coming from an explicit, Zariski dense set, and $X_1,\ldots,X_{d+1}\subseteq\mathbb{F}_q$ are suitably large, then $|P(X_1,\ldots,X_{d+1})|=q-O(1)$. Our methods rely on a higher-degree extension of a result of Vinh on point--line incidences over a finite field. |
| title | Expansion properties of polynomials over finite fields |
| topic | Combinatorics Number Theory 11B06, 11B30, 51B05 |
| url | https://arxiv.org/abs/2403.03732 |