### AIMS Mathematics

2017, Issue 4: 586-609. doi: 10.3934/Math.2017.4.586
Research article

# Permutational behavior of reversed Dickson polynomials over finite fields II

• Received: 06 June 2017 Accepted: 16 October 2017 Published: 06 November 2017
• In this paper, we study the special reversed Dickson polynomial of the form $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$, where $s, e_1, ..., e_s$ are positive integers, $\ell$ is an integer with $0\le \ell < p$. In fact, by using Hermite criterion we first give an answer to the question that the reversed Dickson polynomials of the forms $D_{p^{s}+1, k}(1, x)$, $D_{p^{s}+2, k}(1, x)$, $D_{p^{s}+3, k}(1, x)$, $D_{p^{s}+4, k}(1, x)$, $D_{p^{s}+p^{t}, k}(1, x)$ and $D_{p^{s}+p^{t}+1, k}(1, x)$ are permutation polynomials of ${\mathbb F}_{q}$ or not. Finally, utilizing the recursive formula of the reversed Dickson polynomials, we represent $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$ as the linear combination of the elementary symmetric polynomials with the power of $1-4x$ being the variables. From this, we present a necessary and sufficient condition for $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$ to be a permutation polynomial of ${\mathbb F}_{q}$.

Citation: Kaimin Cheng. Permutational behavior of reversed Dickson polynomials over finite fields II[J]. AIMS Mathematics, 2017, 2(4): 586-609. doi: 10.3934/Math.2017.4.586

### Related Papers:

• In this paper, we study the special reversed Dickson polynomial of the form $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$, where $s, e_1, ..., e_s$ are positive integers, $\ell$ is an integer with $0\le \ell < p$. In fact, by using Hermite criterion we first give an answer to the question that the reversed Dickson polynomials of the forms $D_{p^{s}+1, k}(1, x)$, $D_{p^{s}+2, k}(1, x)$, $D_{p^{s}+3, k}(1, x)$, $D_{p^{s}+4, k}(1, x)$, $D_{p^{s}+p^{t}, k}(1, x)$ and $D_{p^{s}+p^{t}+1, k}(1, x)$ are permutation polynomials of ${\mathbb F}_{q}$ or not. Finally, utilizing the recursive formula of the reversed Dickson polynomials, we represent $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$ as the linear combination of the elementary symmetric polynomials with the power of $1-4x$ being the variables. From this, we present a necessary and sufficient condition for $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1, x)$ to be a permutation polynomial of ${\mathbb F}_{q}$. ###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142 0.882 0.9

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