Extending a result of Carlitz and McConnel to polynomials which are not permutations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908043234508800 |
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| author | Csajbók, Bence |
| author_facet | Csajbók, Bence |
| contents | Let $D$ denote the set of directions determined by the graph of a polynomial $f$ of $\mathbb{F}_q[x]$, where $q$ is a power of the prime $p$. If $D$ is contained in a multiplicative subgroup $M$ of $\mathbb{F}_q^\times$, then by a result of Carlitz and McConnel it follows that $f(x)=ax^{p^k}+b$ for some $k\in \mathbb{N}$. Of course, if $D\subseteq M$, then $0\notin D$ and hence $f$ is a permutation. If we assume the weaker condition $D\subseteq M \cup \{0\}$, then $f$ is not necessarily a permutation, but Sziklai conjectured that $f(x)=ax^{p^k}+b$ follows also in this case. When $q$ is odd, and the index of $M$ is even, then a result of Ball, Blokhuis, Brouwer, Storme and Sz\H onyi combined with a result of McGuire and Göloğlu proves the conjecture. Assume $°f\geq 1$. We prove that if the size of $D^{-1}D=\{d^{-1}d' : d\in D\setminus \{0\},\, d'\in D\}$ is less than $q-°f+2$, then $f$ is a permutation of $\mathbb{F}_q$. We use this result to verify the conjecture of Sziklai. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04045 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extending a result of Carlitz and McConnel to polynomials which are not permutations Csajbók, Bence Combinatorics Number Theory Let $D$ denote the set of directions determined by the graph of a polynomial $f$ of $\mathbb{F}_q[x]$, where $q$ is a power of the prime $p$. If $D$ is contained in a multiplicative subgroup $M$ of $\mathbb{F}_q^\times$, then by a result of Carlitz and McConnel it follows that $f(x)=ax^{p^k}+b$ for some $k\in \mathbb{N}$. Of course, if $D\subseteq M$, then $0\notin D$ and hence $f$ is a permutation. If we assume the weaker condition $D\subseteq M \cup \{0\}$, then $f$ is not necessarily a permutation, but Sziklai conjectured that $f(x)=ax^{p^k}+b$ follows also in this case. When $q$ is odd, and the index of $M$ is even, then a result of Ball, Blokhuis, Brouwer, Storme and Sz\H onyi combined with a result of McGuire and Göloğlu proves the conjecture. Assume $°f\geq 1$. We prove that if the size of $D^{-1}D=\{d^{-1}d' : d\in D\setminus \{0\},\, d'\in D\}$ is less than $q-°f+2$, then $f$ is a permutation of $\mathbb{F}_q$. We use this result to verify the conjecture of Sziklai. |
| title | Extending a result of Carlitz and McConnel to polynomials which are not permutations |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2409.04045 |