Zero-free sector of the Wronski map on the totally nonnegative Grassmannian
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| Format: | Preprint |
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2025
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| _version_ | 1866916965134630912 |
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| author | Karp, Steven N. |
| author_facet | Karp, Steven N. |
| contents | 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 $|\arg(z)| < \fracπ{n}$ of the complex plane, and the bound $\fracπ{n}$ 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 $|\arg(z)| < \fracπ{n}$ (independent of $k$), and the bound $\fracπ{n}$ is tight. Our proof uses classical results of Gantmakher and Krein (1950) and Obreschkoff (1923) on sign variation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18478 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zero-free sector of the Wronski map on the totally nonnegative Grassmannian Karp, Steven N. Classical Analysis and ODEs Combinatorics Complex Variables 30C15, 26C10, 14M15, 14N15, 15B48, 34C10 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 $|\arg(z)| < \fracπ{n}$ of the complex plane, and the bound $\fracπ{n}$ 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 $|\arg(z)| < \fracπ{n}$ (independent of $k$), and the bound $\fracπ{n}$ is tight. Our proof uses classical results of Gantmakher and Krein (1950) and Obreschkoff (1923) on sign variation. |
| title | Zero-free sector of the Wronski map on the totally nonnegative Grassmannian |
| topic | Classical Analysis and ODEs Combinatorics Complex Variables 30C15, 26C10, 14M15, 14N15, 15B48, 34C10 |
| url | https://arxiv.org/abs/2508.18478 |