  Let $M$ be a square matrix and let $p(t)$ be a monic polynomial of degree
$n$. Let $Z$ be a set of $n\times n$ matrices. The multiplicative inverse
eigenvalue problem asks for the construction of a matrix in $Z$ such that the
product matrix $MZ$ has characteristic polynomial $p(t)$. In this paper we
provide new necessary and sufficient conditions when $Z$ is an affine variety
over an algebraically closed field.
