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:

1. K. Cheng, Permutational Behavior of Reversed Dickson Polynomials over Finite Fields, AIMS Math., 2 (2017), 244-259.
2. R. Coulter, Explicit evaluation of some Weil sums, Acta Arith., 83 (1998), 241-251.
3. S. Hong, X. Qin andW. Zhao, Necessary conditions for reversed Dickson polynomials of the second kind to be permutational, Finite Fields Appl., 37 (2016), 54-71.
4. X. Hou, G. L. Mullen, J.A. Sellers and J.L. Yucas, Reversed Dickson polynomials over finite fields, Finite Fields Appl., 15 (2009), 748-773.
5. R. Lidl and H. Niederreiter, Finite Fields, second ed., Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 20, 1997.
6. X. Qin and S. Hong, Constructing permutation polynomials over finite fields, Bull. Aust. Math. Soc., 89 (2014), 420-430.
7. X. Qin, G. Qian and S. Hong, New results on permutation polynomials over finite fields, Int. J. Number Theory, 11 (2015), 437-449.
8. Q. Wang and J. Yucas, Dickson polynomials over finite fields, Finite Fields Appl., 18 (2012), 814-831.

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This article has been cited by:

1. Weihua Li, Chengcheng Fang, Wei Cao, On the number of irreducible polynomials of special kinds in finite fields, AIMS Mathematics, 2020, 5, 4, 2877, 10.3934/math.2020185

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