Numerical Schubert calculus
Birkett Huber, Frank Sottile, Bernd Sturmfels
TL;DR
This work develops semi-numerical, path-continuation methods for solving polynomial systems arising from Schubert calculus on Grassmannians. It introduces two optimal extrinsic deformations for hypersurface Schubert conditions—the Gröbner homotopy (Plücker-ideal) and the SAGBI homotopy (projective coordinate ring)—and a Pieri-based intrinsic deformation (Pieri homotopy) for general special Schubert conditions, all yielding exactly $d$ solution planes without path divergence. The Pieri framework relies on moving cycles, Pieri trees, and Kostka numbers to reduce the problem to transversal triple intersections and sequential homotopies, with correctness grounded in Pieri's formula and Schubert calculus. The methods are demonstrated computationally on moderate-size instances, with applications to control theory (pole assignment) and real enumerative geometry, and are shown to outperform traditional Gröbner-based approaches in several cases. Overall, the paper provides a robust toolkit for numerically solving Schubert-geometry problems via optimal, structure-exploiting homotopies and illustrates practical impact through computational results.
Abstract
We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gröbner basis for the Plücker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.