Zero-free sector of the Wronski map on the totally nonnegative Grassmannian

TL;DR

This paper extends classical zero-free sector results for polynomials with nonnegative coefficients to Wronskians of polynomials associated with the totally nonnegative Grassmannian, establishing a tight bound on the sector where zeros cannot occur.

Contribution

It generalizes the zero-free sector result from polynomials to Wronskians of polynomials in the totally nonnegative Grassmannian, linking classical analysis with modern algebraic geometry.

Findings

The Wronskian of such polynomials has no zeros in the sector | ext{arg}(z)| < rac{ extpi}{n}

The zero-free sector bound rac{ extpi}{n} is proven to be tight

The proof leverages classical results on sign variation by Gantmakher, Krein, and Obreschkoff.

Abstract

A classical result states that if $f(z)$ is a polynomial of degree at most $n$ with nonnegative coefficients, then $f(z)$ has no zeros in the sector $|\arg(z)| < \frac{\pi}{n}$ of the complex plane, and the bound $\frac{\pi}{n}$ is tight. Motivated by the Shapiro--Shapiro conjecture and related problems in real Schubert calculus, we generalize this result to Wronskians of polynomials. Namely, let $f_1(z), \dots, f_k(z)$ be linearly independent polynomials of degree at most $n$ whose coefficient matrix has all nonnegative $k\times k$ minors (that is, the polynomials span an element of the totally nonnegative Grassmannian in the sense of Lusztig and Postnikov). We show that the Wronskian polynomial $\operatorname{Wr}(f_1, \dots, f_k)$ has no complex zeros in the sector $|\arg(z)| < \frac{\pi}{n}$ (independent of $k$), and the bound $\frac{\pi}{n}$ is tight. Our proof uses classical results of Gantmakher and Krein (1950) and Obreschkoff (1923) on sign variation.