arXiv subjects
Publications and source records attributed to Xiaoer Qin.
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 $\sum_{a\in {\mathbb F}_{q}}D_{n,3}(1,a)$.
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 $1\leq r_1<r_2$, $1\leq r_3<r_4$, $1\le n_1<n_2$, $1\le n_3<n_4$, $n_1\leq n_3$, $b_1, b_2\in \mathbb{F}_{q}$, $a_{1i}\in \mathbb{F}_{q}^{*}$ $(1\leq i\leq r_2)$, $a_{2j}\in \mathbb{F}_{q}^{*}$$(1\leq j\leq r_4)$ and the exponent of each variable is a positive integer. An example is also presented to demonstrate the validity of the main result.
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 $L(x)+\sum_{j=1}^k γ_jh_j(f_j(x))$ and $x+\sum_{j=1}^kγ_jf_j(x)$. These generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form $L(x)+\sum_{i=1} ^lγ_i {\rm Tr}_{{\bf F}_{q^m}/{\bf F}_{q}}(h_i(x))$, which extends a theorem of Charpin and Kyureghyan obtained in 2009.
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 $\sum_{a\in \mathbb{F}_{q}}E_{n}(1,a)$.
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 $\sum_{i=1}^k(L_{i}(x)+γ_i)h_i(B(x))$ 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 $x h(λ_{j}(x))$ and $xh(μ_{j}(x))$, where $λ_{j}(x)$ is the $j$-th elementary symmetric polynomial of $x, x^{q}, ..., x^{q^{m-1}}$ and $μ_{j}(x)=\textup{Tr}_{{\bf F}_{q^{m}}/{\bf F}_{q}}(x^{j})$. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form $L_1(x)+L_{2}(γ)h(f(x))$ over ${\bf F}_{q^{m}}$, which extends a result of Kyureghyan.