Lower bounds in real Schubert calculus

Abstract

We describe a large-scale computational experiment studying 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 that their real osculating instances have the observed lower bounds and gaps.