Intersection theory for linear systems

In this paper we explicitly describe, using the the methods of algebraic geometry and topology, a calculus for studying intersections of various classes of linear systems of special interest. For example, to say the class of systems with a fixed set of poles and the class of systems feedback equivalent to a given system has a nonempty intersection is to say we can place those poles via feedback. These intersection rings are described in detail and applied to the pole-assignment problem, for the manifold of all n-dimensional, m-input controllable, linear systems while partial results are given for the parameter space of m-input, p-output linear systems of McMillan degree n.