Permutation Polynomials of $x^{1+\frac{q-1}{m}}+ax$
Received:November 10, 2021  Revised:November 25, 2022
Key Words: polynomial   permutation polynomial   finite field  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12171163) and the Guangdong Basic and Applied Basic Research Foundation (Grant No.2020A1515111090).
Author NameAffiliation
Danyao WU School of Computer Science and Technology, Dongguan University of Technology, Guangdong 523808, P. R. China 
Pingzhi YUAN School of Mathematics, South China Normal University, Guangdong 510631, P. R. China 
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      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 $F_{q^r}~(r\geq2)$.
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