The geometry of the partial realization problem

It is shown that the space of sequences of length n which have an extrapolation of McMillan degree k, and no extrapolations of lower McMillan degree can be given the structure of a differentiable manifold. This approach makes the proof of certain known results on the partial realization problem quite straightforward and makes it possible to establish some important new results as well. A key tool is the fact, proven here, that the set of n by a real Hankel matrices of rank r is a manifold with r+1 connected components.