In this paper, the notions of pseudofield of values and joint pseudofield of values of matrix polynomials are introduced and some of their algebraic and geometrical properties are studied.  Moreover, the relationship between the pseudofield of values of a matrix polynomial and the pseudofield of values of its companion linearization is stated, and then some properties of the augmented field of ...

Glossary Canonical realization: A state-space realization that has specific properties that makes it unique. Characteristic polynomial: The scalar denominator of a transfer function. Column-reduced: A polynomial matrix with non-singular column leading coefficient matrix. Coprime: See relatively prime. Denominator polynomial: The polynomial appearing in the denominator of a rational transfer fun...

Consider an n × <span style="fon...

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions a coefficient by (elementary) matrix-variables over ground ring in not necessary commuting variables. These subalgebras full rings show up noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining precise lift Hilbert Basis Theorem when...

The cover polynomial C(D) = C(D;x, y) of a digraph D is a twovariable polynomial whose coefficients are determined by the number of vertex coverings of D by directed paths and cycles. Just as for the Tutte polynomial for undirected graphs (cf. [11, 16]), various properties of D can be read off from the values of C(D;x, y). For example, for an n-vertex digraph D, C(D; 1, 0) is the number of Hami...