Properties of a generalized Sylvester matrix

We consider problems close to that of the minimal stabilization of a linear vector (i.e., MISO or SIMO) dynamic system; more specifically, the problem of determining the number of common roots of a family of polynomials, and investigating the properties of the so-called generalized Sylvester matrix. The classical definition of the Sylvester matrix is valid for two polynomials, and there are different methods for defining the generalized (extended) Sylvester matrix for a family of polynomials. In this work, we consider a definition of the generalized Sylvester matrix and its properties in the context of their potential future application for solving the minimal stabilization problem.