From $r$-Linearized Polynomial Equations to $r^m$-Linearized Polynomial Equations

Abstract

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 $z\in {\bar{\mathbb{F}}}_r$ satisfies the equation ${\sum }_{i=0}^{m-1}{a}_i{f}_i(z{)}^{r^i}=0$, where ${\sum }_{i=0}^{m-1}{a}_i{f}_i^{r_i}\in {\mathbb{F}}_q[x]$ 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.

11 pages

References in corpus (1)

• Some algorithms for skew polynomials over finite fields