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AIMS Mathematics

2017, Volume 2, Issue 2: 244-259. doi: 10.3934/Math.2017.2.244
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Research article

Permutational behavior of reversed Dickson polynomials over finite fields

  • School of Mathematics and information, China west normal University, Nanchong 637009, P.R. China
  • Received: 15 March 2017 Accepted: 18 April 2017 Published: 20 April 2017

  • In this paper, we develop the method presented previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n, k}(1, x)$ of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n, k}(1, a)$. Our results extend the results of Hong, Qin and Zhao to the general $k\ge 0$ case.

    Citation: Kaimin Cheng. Permutational behavior of reversed Dickson polynomials over finite fields[J]. AIMS Mathematics, 2017, 2(2): 244-259. doi: 10.3934/Math.2017.2.244

    Related Papers:

  • In this paper, we develop the method presented previously by Hong, Qin and Zhao to obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n, k}(1, x)$ of the $(k+1)$-th kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n, k}(1, a)$. Our results extend the results of Hong, Qin and Zhao to the general $k\ge 0$ case.


    加载中
    [1] S. D. Cohen, Dickson polynomials of the second kind that are permutations, Canad. J. Math., 46 (1994), 225-238.
    [2] 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.
    [3] 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.
    [4] R. Lidl and H. Niederreiter, Finite Fields, second ed. , Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 20 1997.
    [5] X. Qin and S. Hong, Constructing permutation polynomials over finite fields, Bull. Aust. Math. Soc., 89 (2014), 420-430.
    [6] X. Qin, G. Qian and S. Hong, New results on permutation polynomials over finite fields, Int. J. Number Theory, 11 (2015), 437-449.
    [7] Q. Wang and J. L. Yucas, Dickson polynomials over finite fields, Finite Fields Appl., 18 (2012), 814-831.
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  • © 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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    沈阳化工大学材料科学与工程学院 沈阳 110142

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