A topological proof of the Shapiro-Shapiro conjecture

TL;DR

This paper provides a topological proof of the Shapiro-Shapiro conjecture, extending it to complex conjugate roots and analyzing the Wronski map's properties through topological and combinatorial methods.

Contribution

It generalizes the conjecture to include complex roots, decomposes the real Schubert cell, and relates the Wronski map's degree to symmetric group characters.

Findings

The Wronski map's degree relates to symmetric group characters.

When all roots are real, the Wronski map is a trivial covering.

Provides a new topological proof of the original conjecture.

Abstract

We prove a generalization of the Shapiro-Shapiro conjecture on Wronskians of polynomials, allowing the Wronskian to have complex conjugate roots. We decompose the real Schubert cell according to the number of real roots of the Wronski map, and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given as an evaluation of a symmetric group character. In the case where all roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.