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In this paper, we obtain several results on the permutational behavior of the reversed Dickson polynomial $D_{n,3}(1,x)$ of the fourth kind over the finite field ${\mathbb F}_{q}$. Particularly, we present the explicit evaluation of the first moment $\sum_{a\in {\mathbb F}_{q}}D_{n,3}(1,a)$., Comment: 16 pages

Let $\mathbb{F}_q$ denote the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ represent the nonzero elements of $\mathbb{F}_{q}$. In this paper, by using the Smith normal form we give an explicit formula for the number of rational points of the algebraic variety defined by the following system of equations over $\mathbb{F}_{q}$: \begin{align*} {\left\{\begin{array}{rl} &\sum_{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} ...x_{n_1}^{e^{(1)}_{i,n_1}} +\sum_{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}} ...x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0,\\ &\sum_{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}} ...x_{n_3}^{e^{(2)}_{j,n_3}} +\sum_{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}} ...x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, \end{array}\right.} \end{align*} where the integers $1\leq r_1

In this paper, we present several necessary conditions for the reversed Dickson polynomial $E_{n}(1, x)$ of the second kind to be a permutation of $\mathbb{F}_{q}$. In particular, we give explicit evaluation of the sum $\sum_{a\in \mathbb{F}_{q}}E_{n}(1,a)$., Comment: 13 pages

In this paper, we get several new results on permutation polynomials over finite fields. First, by using the linear translator, we construct permutation polynomials of the forms $L(x)+\sum_{j=1}^k \gamma_jh_j(f_j(x))$ and $x+\sum_{j=1}^k\gamma_jf_j(x)$. These generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form $L(x)+\sum_{i=1} ^l\gamma_i {\rm Tr}_{{\bf F}_{q^m}/{\bf F}_{q}}(h_i(x))$, which extends a theorem of Charpin and Kyureghyan obtained in 2009., Comment: 11 pages. To appear in International Journal of Number Theory

In this paper, we construct several new permutation polynomials over finite fields. First, using the linearized polynomials, we construct the permutation polynomial of the form $\sum_{i=1}^k(L_{i}(x)+\gamma_i)h_i(B(x))$ over ${\bf F}_{q^{m}}$, where $L_i(x)$ and $B(x)$ are linearized polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalize a result of Marcos by constructing permutation polynomials of the forms $x h(\lambda_{j}(x))$ and $xh(\mu_{j}(x))$, where $\lambda_{j}(x)$ is the $j$-th elementary symmetric polynomial of $x, x^{q}, ..., x^{q^{m-1}}$ and $\mu_{j}(x)=\textup{Tr}_{{\bf F}_{q^{m}}/{\bf F}_{q}}(x^{j})$. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form $L_1(x)+L_{2}(\gamma)h(f(x))$ over ${\bf F}_{q^{m}}$, which extends a result of Kyureghyan., Comment: 9 pages. To appear in Bulletin of the Australian Mathematical Society