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Permutational behavior of reversed Dickson polynomials over finite fields II

Kaimin Cheng

*Corresponding author: Kaimin Cheng ckm20@126.com


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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