  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.
