The monotone secant conjecture in the real Schubert calculus

TL;DR

This paper investigates the monotone secant conjecture in real Schubert calculus, providing theoretical and computational evidence that a broad class of polynomial systems derived from flag manifolds have all real solutions.

Contribution

It introduces the monotone secant conjecture as a generalization of the Shapiro conjecture and offers both theoretical insights and extensive computational data supporting it.

Findings

The conjecture holds for various cases based on computational evidence.

Observed phenomena suggest patterns in the real solutions of these polynomial systems.

Theoretical evidence aligns with computational results, supporting the conjecture's validity.

Abstract

The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 teraHertz-years of computing, and we discuss some of the phenomena we observed in our data.