On the Degree of Mixed Polynomial Matrices

The mixed polynomial matrix, introduced as a convenient mathematical tool for the description of physical/engineering dynamical systems, is a polynomial matrix of which the coefficients are classified into fixed constants and independent parameters. The valuated matroid, invented by Dress and Wenzel [Appl. Math. Lett., 3 (1990), pp. 33–35], is a combinatorial abstraction of the degree of minors (subdeterminants) of a polynomial matrix. We discuss a number of implications of the recent developments in the theory of valuated matroids in the context of polynomial matrix theory. In particular, we apply the valuated matroid intersection theorem to the analysis of the degree of the determinant of a mixed polynomial matrix to obtain a novel duality identity together with an efficient algorithm.