标题： Wronski映射在完全非负格拉斯曼流形上的无零区域 显示英文标题

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

摘要： 一个经典的结果指出，如果$f(z)$是一个次数最多为$n$且系数非负的多项式，那么$f(z)$在复平面上的扇形区域$|\arg(z)| < \frac{\pi}{n}$中没有零点，且界$\frac{\pi}{n}$是紧的。 受到 Shapiro--Shapiro 猜想及相关实 Schubert 微积分问题的启发，我们将这一结果推广到多项式的 Wronskians。 即，设 $f_1(z), \dots, f_k(z)$ 为次数不超过 $n$ 的线性无关多项式，其系数矩阵的所有非负 $k\times k$ 子式（即这些多项式张成一个在Lusztig和Postnikov意义下的完全非负Grassmannian的元素）。我们证明Wronskian多项式 $\operatorname{Wr}(f_1, \dots, f_k)$ 在扇形 $|\arg(z)| < \frac{\pi}{n}$ 中没有复数零点（与 $k$ 无关），并且界限 $\frac{\pi}{n}$ 是紧的。我们的证明使用了Gantmakher和Krein（1950）以及Obreschkoff（1923）关于符号变化的经典结果。 显示英文摘要

摘要： 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.