Completion of quasi-regular matrix pencils

In this paper we give solutions for two important particular cases of the general completion problem, involving quasi-regular matrix pencil. We describe the possible Kronecker invariants of a quasi-regular pencil whose subpencil is prescribed in the case when the prescribed subpencil is without nontrivial homogeneous invariant factors. Also, we solve “the dual problem” by describing the possible Kronecker invariants of a pencil without nontrivial homogeneous invariant factors when a quasi-regular subpencil is prescribed. These two results are expected to be the crucial steps toward a solution of the general completion problem. The results are given over algebraically closed fields.