Certifiable Numerical Computations in Schubert Calculus

TL;DR

This paper introduces a primal-dual formulation for Schubert calculus problems that transforms complex polynomial systems into bilinear equations, enabling reliable numerical certification of solutions.

Contribution

It proposes a novel primal-dual approach using dual Grassmannian parametrizations to convert Schubert calculus problems into complete intersections of bilinear equations.

Findings

Enables numerical certification of Schubert calculus problems.

Transforms polynomial systems into bilinear equations.

Facilitates more reliable computational solutions.

Abstract

Traditional formulations of geometric problems from the Schubert calculus, either in Plucker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that are typically far from complete intersections and (in local coordinates) typically of degree exceeding two. We present an alternative primal-dual formulation using parametrizations of Schubert cells in the dual Grassmannians in which intersections of Schubert varieties become complete intersections of bilinear equations. This formulation enables the numerical certification of problems in the Schubert calculus.