Injectivity Properties of Generalized Wronski Map

Given a complex vector space V, consider the quotient map of the image of the Plucker embedding of the Grassmannian of m-planes of V by a certain subspace of P ∧m V . Such maps generalize the classical Wronski maps on Grassmannians of spaces of polynomials, the Wronski maps on Grassmanians of spaces of solutions of linear homogeneous differential equations, pole-placement maps of input-output linear systems and their realizations as linear control systems. We are interested in finding the degree of such maps, i.e. in determining the number of points in the preimage of the generic point of the image. We distinguish a special subclass of these maps, called self-adjoint, for which the degree of the corresponding Wronski map is at least two. In the case of Wronski maps on Grassmanians of spaces of solutions of linear homogeneous differential equations our self-adjoint generalized Wronski maps correspond to the classical self-adjoint linear differential operators, up to a natural equivalence. In the case of linear control systems, they correspond to control system with symmetric transfer function, up to a state-feedback equivalence. The main question is whether there are non-selfadjoint generalized Wronski maps with the degree greater than 1. We give a negative answer to this question in the case m = 2 and m = 3 under some natural assumptions