A strengthening of McConnel's theorem on permutations over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915148639240192 |
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| author | Yip, Chi Hoi |
| author_facet | Yip, Chi Hoi |
| contents | Let $p$ be a prime, $q=p^n$, and $D \subset \mathbb{F}_q^*$. A celebrated result of McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^*$, and $f:\mathbb{F}_q \to \mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on $D$ to obtain the same conclusion on $f$. In particular, we show that McConnel's theorem extends if $D$ has small doubling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_21362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A strengthening of McConnel's theorem on permutations over finite fields Yip, Chi Hoi Number Theory Combinatorics 11T06, 11B30 Let $p$ be a prime, $q=p^n$, and $D \subset \mathbb{F}_q^*$. A celebrated result of McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^*$, and $f:\mathbb{F}_q \to \mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on $D$ to obtain the same conclusion on $f$. In particular, we show that McConnel's theorem extends if $D$ has small doubling. |
| title | A strengthening of McConnel's theorem on permutations over finite fields |
| topic | Number Theory Combinatorics 11T06, 11B30 |
| url | https://arxiv.org/abs/2407.21362 |