The multiplicative inverse eigenvalue problem over an algebraically closed field

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