Coordinate avoidance from a nonzero permanent
= Coordinate avoidance from a nonzero permanent
If an $n$ by $n$ matrix $A$ over a field has nonzero <permanent of a matrix>[permanent], $b\in F^n$, and every $S_i\subseteq F$ has two elements, then some $x\in\prod_iS_i$ makes every coordinate of $Ax-b$ nonzero. Apply the <Combinatorial Nullstellensatz> to
$$
\prod_{i=1}^n((Ax)_i-b_i),
$$
whose coefficient of $x_1\cdots x_n$ is $\operatorname{perm}A$.