In group theory, various properties of the abelianization of a cyclically presented group can be deduced by the Smith normal form of an integer circulant matrix. Motivated by this fact, we present a number of results on the Smith form of matrices that are polynomials in the companion matrix of a polynomial with coefficients in a generic elementary divisor domain. Our tools are of purely matrix theoretical nature and I will present them from the point of view of a matrix theorist. However, they enable significant advances in pure algebra. In particular, I plan to discuss how our results provide a tool to study the homology of 3-dimensional Brieskorn manifolds, which are objects of (apparently) interest for topologists. If time permits, I will mention other possibile applications to group theory.