Pieri Homotopies for Problems in Enumerative Geometry
applied to Pole Placement in Linear Systems Control

Birkett Huber and Jan Verschelde

Abstract:

In the numerical Schubert calculus paper, Pieri homotopies were proposed to enumerate all $p$-planes in $\cc^{m+p}$ that meet $n$ given $(m+1-k_i)$-planes in general position, with $k_1+k_2+\cdots+k_n = mp$ as condition to have a finite number of solution $p$-planes. Essentially the 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. In this paper 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. It also provides a self-contained proof of correctness. We were also able to extend our approach to the quantum Schubert calculus problem of enumerating all polynomial maps of degree $q$ into the Grassmannian of $p$-planes in $\cc^{m+p}$ 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.

AMS Subject Classification : 14N10, 14M15, 65H10, 68Q40, 93B27, 93B55.

keywords : polynomial system, continuation methods, numerical Schubert calculus, Pieri homotopy, cheater's homotopy, dynamic pole placement problem, control theory, linear system, quantum Schubert calculus, enumerative geometry, Grassmannian.

SIAM Journal on Control and Optimization 38(4): 1265-1287, 2000.