# Finite and Infinite Elementary Divisors of Matrix Polynomials: A Global Approach

Zaballa, Ion and Tisseur, Francoise (2012) Finite and Infinite Elementary Divisors of Matrix Polynomials: A Global Approach. [MIMS Preprint]

There is general agreement on the definition of the finite elementary divisors of a matrix polynomial $Q(\l)\in\F[\l]^{m\times n}$, where $\F$ an arbitrary field. Regarding the elementary divisors at infinity, or infinite elementary divisors, such an agreement has not been so unanimous. We define the infinite elementary divisors of $Q(\l)$ to be the elementary divisors of $\l^\ell Q(\l^{-1})$ at 0, where $\ell$ is the degree of $Q(\l)$. We show that this is the most natural definition if one applies the usual geometric technique of using homogeneous coordinates to deal with the point at infinity. We call our approach global because the homogeneous invariant factors of $Q(\l)$ are defined for all points of the projective line and to distinguish it from another possible approach that, using local rings, leads to the same conclusions.