Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control

Huber, Sottile, and Sturmfels [J. Symbolic Comput., 26 (1998), pp. 767-788] proposed Pieri homotopies to enumerate all p-planes in Cm+p that meet n given (m + 1 - k(i))-planes in general position, with k(1) + k(2) + ... + k(n) = mp as a condition to have a finite number of solution p-planes. Pieri homotopies turn the deformation arguments of classical Schubert calculus into effective numerical methods by expressing the deformations algebraically and applying numerical path-following techniques. We describe the Pieri homotopy algorithm in terms of a poset of simpler problems. This approach is more intuitive and more suitable for computer implementation than the original chain-oriented description and provides also a self-contained proof of correctness. We extend the Pieri homotopies to the quantum Schubert calculus problem of enumerating all polynomial maps of degree q into the Grassmannian of p-planes in Cm+p that meet mp + q ( m + p) given m-planes in general position sampled at mp + q ( m + p) interpolation points. Our approach mirrors existing counting methods for this problem and yields a numerical implementation for the dynamic pole placement problem in the control of linear systems.