Complete Resolution of B.Shapiro's Conjecture 12
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909833865723904 |
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| author | Ma, Lande Ma, Zhaokun |
| author_facet | Ma, Lande Ma, Zhaokun |
| contents | For any real polynomial $p(x)$ of even degree $n$, Shapiro [{\it Arnold Math. J.} 1(1) (2015), 91--99] conjectured that the sum of the number of real zeros of $(n-1)(p')^2 - np p''$ and the number of real zeros of $p$ is positive. We resolve this conjecture completely: it holds in nine mutually exclusive cases and fails in four, as characterized by the root locus properties of general real rational functions. Our results provide a complete classification of real polynomials of even degree with respect to this conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_08957 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete Resolution of B.Shapiro's Conjecture 12 Ma, Lande Ma, Zhaokun Complex Variables 30C10, 30C15, 26C10, 93C05 For any real polynomial $p(x)$ of even degree $n$, Shapiro [{\it Arnold Math. J.} 1(1) (2015), 91--99] conjectured that the sum of the number of real zeros of $(n-1)(p')^2 - np p''$ and the number of real zeros of $p$ is positive. We resolve this conjecture completely: it holds in nine mutually exclusive cases and fails in four, as characterized by the root locus properties of general real rational functions. Our results provide a complete classification of real polynomials of even degree with respect to this conjecture. |
| title | Complete Resolution of B.Shapiro's Conjecture 12 |
| topic | Complex Variables 30C10, 30C15, 26C10, 93C05 |
| url | https://arxiv.org/abs/2510.08957 |