Permutation Polynomials of $x^{1+\frac{q-1}{m}}+ax$

Abstract:

      Let $m$ be a positive integer and $F_{q^r}$ be a finite field with the characteristic of $p$. We prove that if $p>m^2-m$ and $q\equiv 1\pmod{m}$, the polynomial $x^{1+\frac{q-1}{m}}+ax~(a\neq0)$ is not a permutation polynomial over $F_{q^r}~(r\geq2)$. And we verify that if $q\equiv 1\pmod{7}$ and $p\neq 2, 3$, then the polynomial $x^{1+\frac{q-1}{7}}+ax~(a\neq0)$ is not a permutation polynomial over $F_{q^r}~(r\geq2)$.