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

TL;DR

This paper proves that solutions to certain $r$-linearized polynomial equations over finite fields also satisfy related $q$-linearized polynomial equations, explaining many permutation polynomials found via computational methods.

Contribution

It establishes a general result linking solutions of $r$-linearized equations to $q$-linearized equations, enhancing understanding of permutation polynomials.

Findings

Solutions satisfy $q$-linearized polynomial equations with coefficients in $F_r$

Provides theoretical explanation for previously discovered permutation polynomials

Connects $r$-linearized and $q$-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 $z\in \mathbb{\bar{F}}_r$ 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.