A primal-dual formulation for certifiable computations in Schubert calculus

TL;DR

The paper introduces a primal-dual bilinear formulation for Schubert problems that balances equations and variables, enabling certified numerical solutions in Schubert calculus.

Contribution

It presents a novel primal-dual approach to reformulate Schubert problems as square systems, facilitating certification of numerical solutions.

Findings

Formulation as bilinear equations with equal number of equations and variables

Enables certification of numerical solutions using Smale's lpha-theory

Applicable to Grassmannian and flag manifold Schubert problems

Abstract

Formulating a Schubert problem as the solutions to a system of equations in either Pl\"ucker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale's \alpha-theory.