Solving schubert problems with Littlewood-Richardson homotopies
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Parallel homotopy algorithms to solve polynomial systems
ICMS'06 Proceedings of the Second international conference on Mathematical Software
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Huber, Sottile, and Sturmfels [ J. Symbolic Comput., 26 (1998), pp. 767--788] proposed Pieri homotopies to enumerate all p-planes in %$ç^{m+p}$ ${\mathbb{C}}^{m+p}$ that meet n given (m+1-ki)-planes in general position, with k1+k2+ ... + kn = 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 ${\mathbb{C}}^{m+p}$ 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.