Reality and Computation in Schubert Calculus

TL;DR

This paper advances real osculating Schubert calculus by computationally analyzing over 300 million instances, uncovering new conjectures, proving some, and improving solution methods for Schubert problems.

Contribution

It generalizes the Mukhin-Tarasov-Varchenko Theorem, develops efficient computational techniques, and establishes new invariants and bounds in real Schubert calculus.

Findings

Number of real solutions is congruent modulo four to the total solutions.

Discovered lower bounds for real solutions depending on real osculation points.

Proved several conjectures relating to real solutions in Schubert problems.

Abstract

The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational normal curve at real points. This sparked interest in real osculating Schubert calculus, and computations played a large role in developing the surrounding theory. We uncover generalizations of the Mukhin-Tarasov-Varchenko Theorem, proving them when possible. We also improve the state of the art of computationally solving Schubert problems, allowing us to more effectively study ill-understood phenomena in Schubert calculus. We use supercomputers to methodically solve real osculating instances of Schubert problems. By studying over 300 million instances of over 700 Schubert problems, we amass data significant enough to reveal possible generalizations of the Mukhin-Tarasov-Varchenko Theorem and compelling enough to support our conjectures. Combining algebraic geometry and combinatorics, we prove some of these conjectures. To improve the efficiency of solving Schubert problems, we reformulate an instance of a Schubert problem as the solution set to a square system of equations in a higher-dimensional space. During our investigation, we found the number of real solutions to an instance of a symmetrically defined Schubert problem is congruent modulo four to the number of complex solutions. We prove this congruence, giving a new invariant in enumerative real algebraic geometry. We also discovered a family of Schubert problems whose number of real solutions to a real osculating instance has a lower bound depending only on the number of defining flags with real osculation points.