Permutational behavior of reversed Dickson polynomials over finite fields II

Kaimin Cheng

doi:10.3934/Math.2017.4.586

AIMS Mathematics, 2017, 2(4): 586-609. doi: 10.3934/Math.2017.4.586. Research article

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Permutational behavior of reversed Dickson polynomials over finite fields II

Kaimin Cheng

School of Mathematics and Information, China West Normal University, Nanchong 637009, P.R.China

Received: , Accepted: , Published:

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In this paper, we study the special reversed Dickson polynomial of theform $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1,x)$, where $s,e_1, ..., e_s$are positive integers, $\ell$ is an integer with $0 ≤ \ell < p$. In fact, by using Hermite criterion we first give an answer to the questionthat 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 permutationpolynomials of ${\mathbb F}_{q}$ or not. Finally, utilizing the recursiveformula of the reversed Dickson polynomials, we represent$D_{p^{e_1}+...+p^{e_s}+\ell, k}(1,x)$ as the linear combinationof the elementary symmetric polynomials with the power of $1-4x$being the variables. From this, we present a necessary and sufficient conditionfor $D_{p^{e_1}+...+p^{e_s}+\ell, k}(1,x)$ to be a permutation polynomialof ${\mathbb F}_{q}$.

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Keywords: Permutation polynomial; Reversed Dickson polynomial of (k + 1)-th kind; Hermite’s Criterion

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

References:

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Copyright Info: 2017, Kaimin Cheng, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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