Some results on permutation polynomials over finite fields

Permutation polynomials is a hot topic in finite fields. Permutation polynomials with the form $x^rh(x^{q-1})$ of ${\bf F}_{q^{2}}$ have been studied extensively. In this paper, we build a general relation between permutation polynomials having the form $x^rh(x^{q-1})$ over ${\bf F}_{q^{2}}$ and permutation rational functions over ${\bf F}_{q}$. For $p=11, 13$ and $q=p^k$, by using the exceptional polynomials of degrees 11 and 13 of ${\bf F}_{q}$, we focus on studying the permutation properties of trinomials $x^{q+10}(ax^{10(q-1)}+x^{9(q-1)}+1)$ and $x^{q+6}(ax^{9(q-1)}+x^{5(q-1)}+1), x^{q+12}(ax^{12(q-1)}+x^{11(q-1)}+1)$ over ${\bf F}_{q^{2}}$. Conversely, we show an approach to using permutation polynomials over ${\bf F}_{q}$ to get permutation polynomials over ${\bf F}_{q^2}$.