置换多项式$x^{1+\frac{q-1}{m}}+ax$

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

中文关键词:  多项式  置换多项式  有限域

英文关键词:polynomial  permutation polynomial  finite field

中文摘要:

      设$m$为正整数, $F_{q^r}$是特征为$p$的有限域. 本文证明了如果$p>m^2-m$且$q\equiv 1\pmod{m}$, 则多项式$x^{1+\frac{q-1}{m}}+ax~(a\neq0)$不是$F_{q^r}~(r\geq2)$上的置换多项式. 本文还证明了$q\equiv 1\pmod{7}$且$p\neq 2, 3$时, $x^{1+\frac{q-1}{7}}+ax~(a\neq0)$不是$F_{q^r}~(r\geq2)$上的置换多项式

英文摘要:

      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 $F_{q^r}~(r\geq2)$.