On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914819466067968 |
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| author | Katkova, Olga Tyaglov, Mikhail Vishnyakova, Anna |
| author_facet | Katkova, Olga Tyaglov, Mikhail Vishnyakova, Anna |
| contents | For a given real polynomial $p$ we study the possible number of real roots of a differential polynomial $H_{\varkappa}[p](x) = \varkappa\left(p'(x)\right)^2-p(x)p''(x), \varkappa \in \mathbb{R}.$ In the special case when all real zeros of the polynomial $p$ are simple, and all roots of its derivative $p'$ are real and simple, the distribution of zeros of $H_{\varkappa}[p]$ is completely described for each real $\varkappa.$ We also provide counterexamples to two Boris Shapiro's conjectures about the number of zeros of the function $H_{\frac{n-1}{n}}[p].$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture Katkova, Olga Tyaglov, Mikhail Vishnyakova, Anna Complex Variables 26C10, 30C15 For a given real polynomial $p$ we study the possible number of real roots of a differential polynomial $H_{\varkappa}[p](x) = \varkappa\left(p'(x)\right)^2-p(x)p''(x), \varkappa \in \mathbb{R}.$ In the special case when all real zeros of the polynomial $p$ are simple, and all roots of its derivative $p'$ are real and simple, the distribution of zeros of $H_{\varkappa}[p]$ is completely described for each real $\varkappa.$ We also provide counterexamples to two Boris Shapiro's conjectures about the number of zeros of the function $H_{\frac{n-1}{n}}[p].$ |
| title | On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture |
| topic | Complex Variables 26C10, 30C15 |
| url | https://arxiv.org/abs/2406.00686 |