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.
Categories: math.NT