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

Research article

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

Permutational behavior of reversed Dickson polynomials over finite fields II

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

## Abstract    Full Text(HTML)    Figure/Table

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 ${\mathbb F}_{q}$.
Figure/Table
Supplementary
Article Metrics

# 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.