Hamiltonian indices and rational spectral densities

In this note we announce several (global) topological properties of various spaces of linear systems, particularly symmetric, lossless, and Hamiltonian systems and (for the first time) multivariable spectral densities of fixed McMillan degree. Such a study is motivated by a result asserting that on a connected but not simply connected manifold, one cannot find a vector field having a sink as its only critical point. This is illustrated in the scalar case by showing that only on the space of McMillan degree= |Cauchy index| = n, scalar transfer functions can one define a globally convergent vector field. This result holds in discrete-time, as well as for the nonautonomous case. With these motivations in mind, theorems of Bochner and Fogarty are used to show that spaces of transfer functions defined by symmetry conditions are, in fact, smooth algebraic manifolds. In addition, a particularly simple proof of a regularity theorem of J.M.C. Clark is given based on the Bochmer-Fogarty result. We describe a general philosophy for enumerating the path-components of such a manifold, based on earlier work of Brockett, and illustrate this method in the examples indicated above. As a corollary to these illustrations, we obtain the multivariable Hermite-Hurwitz theorem of Anderson-Bitmead as well as analogues, of this identity of the alternating index of a lossless system and the Hamiltonian index of a symmetric Hamiltonian system.