All,
Suppose $A \in Mat(n, \mathbb{F}_{q})$ for $q$ prime, $q \geq 5$, and $n \geq 2^{q-2}$ . Let $\pi_A(x)$ be the permanental polynomial for $A$. That is,
\begin{equation*}
\pi_{A}(x)=per(xI-A).
\end{equation*}
My question is this: if $x=0$ is a root of $\pi_A$ of multiplicity at least $q-2$, does this imply that $A$ is not invertible?
I am working on a problem recreationally and I am unable to proceed without the answer to the above question being in the affirmative (intuitively, I believe this to be the case). However, I have found no useful literature on the subject of permanental roots of matrices over finite fields, and I am at a loss as to how to progress. Any insight is greatly appreciated.