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

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 $\mathrm{\mid }\arg (z)\mathrm{\mid }<\frac{\pi }{n}$ 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 $\mathrm{Wr}(f_1,\dots ,f_k)$ has no complex zeros in the sector $\mathrm{\mid }\arg (z)\mathrm{\mid }<\frac{\pi }{n}$ (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.