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

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  • 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 ${\mathbb F}_{q}$.
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    [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.

    © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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