Abstract
This paper shows the discrete Legendre duality between the two linear algebraic characteristics of polynomial matrices, degrees of subdeterminants and ranks of expanded matrices. The duality also holds between their combinatorial counterparts in graph theory, which serve as upper bounds on the corresponding linear algebraic quantities. Tightness of one of the combinatorial bounds is shown to be equivalent to that of the other. These results extend the recent results for matrix pencils obtained by Murota, and have applications to combinatorial analysis of the Smith-McMillan form at infinity of a rational function matrix.
