Further Investigation on Cyclotomic Mapping Permutation Polynomials over Finite Fields

We explore the connection between cyclotomic mapping permutation polynomials and permutation polynomials of the form $x^{r}f(x^{\frac{q-1}{l}})$ over finite fields. We present a new necessary and a new sufficient condition to verify permutation behavior of such polynomials over finite field. As its application, for particular values of $r$, we point out some permutation trinomials of the form $P(x)=2x^{r+8}+x^{r+4}+2x^r\in {\mathbb{F}}_{13}[x]$, and work on few classes of permutation binomials.