Lower Bounds in Real Schubert Calculus

TL;DR

This paper reports on a large-scale computational study of real solutions in Schubert calculus, revealing bounds, gaps, and congruences, and proves lower bounds for specific families of problems.

Contribution

It introduces a computational approach to analyze real solutions in Schubert problems and proves lower bounds for a family of these problems.

Findings

Identified nontrivial upper and lower bounds for real solutions

Discovered gaps and a congruence modulo four in solution counts

Proved lower bounds for a family of osculating Schubert problems

Abstract

We describe a large-scale computational experiment to study structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems whose computed numbers of real solutions variously exhibit nontrivial upper bounds, lower bounds, gaps, and a congruence modulo four. We present a family of Schubert problems, one in each Grassmannian, and prove their osculating instances have the observed lower bounds and gaps.