Dynamic Polynomial Combinants and Generalised Resultants

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 assign- ment problems leads to the emergence of dynamic polynomial combinants. This paper aims to demonstrate the origin of dynamic polynomial combinants from Linear Systems, and develop the fundamentals of the relevant theory by estab- lishing their link to the theory of Generalised Resultants and examining issues of their parameterization according to the notions of order and degree. The paper provides a description of the key spectral assignment problems, derives the con- ditions for arbitrary assignability of spectrum and introduces a parameterization of combinants according to their order and degree.