A congruence modulo four in real Schubert calculus

TL;DR

This paper proves a new congruence modulo four in real Schubert calculus, extending the known modulo two results, and provides examples where fibers contain real points despite having zero topological degree.

Contribution

It introduces a congruence modulo four in real Schubert calculus and generalizes it to symmetric Schubert problems, strengthening existing results.

Findings

Establishes a congruence modulo four for fibers of the Wronski map.

Generalizes the congruence to symmetric Schubert problems.

Provides examples with real points despite zero topological degree.

Abstract

We establish a congruence modulo four in the real Schubert calculus on the Grassmannian of m-planes in 2m-space. This congruence holds for fibers of the Wronski map and a generalization to what we call symmetric Schubert problems. This strengthens the usual congruence modulo two for numbers of real solutions to geometric problems. It also gives examples of geometric problems given by fibers of a map whose topological degree is zero but where each fiber contains real points.