A lifted square formulation for certifiable Schubert calculus

TL;DR

This paper introduces a new square formulation for Schubert calculus problems by lifting incidence conditions, reducing the number of equations and variables, which aids in certifying numerical solutions with Smale's -theory.

Contribution

It presents a novel lifted square formulation for Schubert problems that is more efficient and suitable for certification compared to previous methods.

Findings

Fewer equations and variables in the new formulation

Enhanced suitability for numerical certification

Improved efficiency in solving 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 usually involves more equations than variables. Using reduction to the diagonal, we previously gave a primal-dual formulation for Schubert problems that involved the same number of variables as equations (a square formulation). Here, we give a different square formulation by lifting incidence conditions which typically involves fewer equations and variables. Our motivation is certification of numerical computation using Smale's \alpha-theory.