Dynamic Polynomial Combinants and Generalised Resultants: Parameterization according to Order and Degree

The theory of constant polynomial combinants has been well developed and it is linked to the linear part of the constant Determinantal Assignment problem that provides the unifying description of the pole and zero assignment problems in Linear Systems. Considering the case of dynamic pole, zero assignment problems leads to the emergence of dynamic polynomial combinants. This paper aims to develop the fundamentals of the theory of polynomial combinants by examining issues of their parameterization of dynamic polynomial combinants according to the notions of order and degree. Central to this study is the link of dynamic combinants to the theory of “Generalised Resultants”. The paper provides a description of the key spectral assignment problems, derives the conditions for arbitrary assignability of spectrum and introduces a complete parameterization of combinants and respective Generalised Resultants which is crucial for studying the minimal degree and order spectrum assignability.