Permutation polynomials of the form $x+γ\mathrm{Tr}(H(x))$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909670033063936 |
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| author | Li, Yangcheng Pang, Xuan Yuan, Pingzhi Zeng, Yuanpeng |
| author_facet | Li, Yangcheng Pang, Xuan Yuan, Pingzhi Zeng, Yuanpeng |
| contents | Given a polynomial \( H(x) \) over \(\mathbb{F}_{q^n}\), we study permutation polynomials of the form \( x + γ\mathrm{Tr}(H(x)) \) over \(\mathbb{F}_{q^n}\). Let \[P_H=\{γ\in \mathbb{F}_{q^n} : x+γ\mathrm{Tr}(H(x))~\text{is a permutation polynomial}\}.\] We present some properties of the set \(P_H\), particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set \(P_H\) and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set \(P_H\) reaches this upper bound, the function \(\mathrm{Tr}(H(x))\) must be an \(\mathbb{F}_q\)-linear function. Finally, we study two classes of functions $H(x)$ over \(\mathbb{F}_{q^2}\) and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permutation polynomials of the form $x+γ\mathrm{Tr}(H(x))$ Li, Yangcheng Pang, Xuan Yuan, Pingzhi Zeng, Yuanpeng Number Theory 11T06, 11T55 Given a polynomial \( H(x) \) over \(\mathbb{F}_{q^n}\), we study permutation polynomials of the form \( x + γ\mathrm{Tr}(H(x)) \) over \(\mathbb{F}_{q^n}\). Let \[P_H=\{γ\in \mathbb{F}_{q^n} : x+γ\mathrm{Tr}(H(x))~\text{is a permutation polynomial}\}.\] We present some properties of the set \(P_H\), particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set \(P_H\) and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set \(P_H\) reaches this upper bound, the function \(\mathrm{Tr}(H(x))\) must be an \(\mathbb{F}_q\)-linear function. Finally, we study two classes of functions $H(x)$ over \(\mathbb{F}_{q^2}\) and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case. |
| title | Permutation polynomials of the form $x+γ\mathrm{Tr}(H(x))$ |
| topic | Number Theory 11T06, 11T55 |
| url | https://arxiv.org/abs/2507.00781 |