Reversed Dickson polynomials of the $(k+1)$-th kind over finite fields, II
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866917878978052096 |
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| author | Fernando, Neranga |
| author_facet | Fernando, Neranga |
| contents | Let $p$ be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$, where $l_1, l_2$, $l_3$, and $l_4$ are non-negative integers. A generalization to $n=p^{l_1}+p^{l_2}+\cdots +p^{l_i}$ is also shown. We find some conditions under which $D_{n,k}(1,x)$ is not a permutation polynomial over finite fields for certain values of $n$ and $k$. We also present a generalization of a recent result regarding $D_{p^l-1,1}(1,x)$ and present some algebraic and arithmetic properties of $D_{n,k}(1,x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1706_01391 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Reversed Dickson polynomials of the $(k+1)$-th kind over finite fields, II Fernando, Neranga Number Theory 11T06, 11T55 Let $p$ be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$, where $l_1, l_2$, $l_3$, and $l_4$ are non-negative integers. A generalization to $n=p^{l_1}+p^{l_2}+\cdots +p^{l_i}$ is also shown. We find some conditions under which $D_{n,k}(1,x)$ is not a permutation polynomial over finite fields for certain values of $n$ and $k$. We also present a generalization of a recent result regarding $D_{p^l-1,1}(1,x)$ and present some algebraic and arithmetic properties of $D_{n,k}(1,x)$. |
| title | Reversed Dickson polynomials of the $(k+1)$-th kind over finite fields, II |
| topic | Number Theory 11T06, 11T55 |
| url | https://arxiv.org/abs/1706.01391 |