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

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Permutational behavior of reversed Dickson polynomials over finite fields II

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

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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