从$r$线性化多项式方程到$r^m$线性化多项式方程
From $r$-Linearized Polynomial Equations to $r^m$-Linearized Polynomial Equations
摘要：
设$r$是素数幂，$q=r^m$。对于$0\le i\le m-1$，令$f_i\in \mathbb{F}_r[x]$为$q$线性化和$a_i\in \mathbb{F}_q$。假设$z\in \mathbb{\bar{F}}_r$满足等式$\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$，其中$\sum_{i=0}^{m-1}a_if_i^{r_i}\in \mathbb{F}_q[x]$是$r$线性化的多项式。证明了$z$满足一个系数在$\mathbb{F}_r$中的$q$线性化多项式方程.这一结果为以前通过计算机搜索得到的众多置换多项式提供了一种解释。
英文摘要：
Let $r$ be a prime power and $q=r^m$. For $0\le i\le m-1$, let $f_i\in \mathbb{F}_r[x]$ be $q$-linearized and $a_i\in \mathbb{F}_q$. Assume that $z\in \mathbb{\bar{F}}_r$ satisfies the equation $\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where $\sum_{i=0}^{m-1}a_if_i^{r_i}\in \mathbb{F}_q[x]$ is an $r$-linearized polynomial. It is shown that $z$ satisfies a $q$-linearized polynomial equation with coefficients in $\mathbb{F}_r$. This result provides an explanation for numerous permutation polynomials previously obtained through computer search.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
