On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities
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| Format: | Preprint |
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2019
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| _version_ | 1866909663251922944 |
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| author | Tyaglov, Mikhail Atia, Mohamed J. |
| author_facet | Tyaglov, Mikhail Atia, Mohamed J. |
| contents | Given a real polynomial $p$ with only real zeroes, we find upper and lower bounds for the number of non-real zeroes of the differential polynomial $$ F_{\varkappa}[p](z):= p(z)p''(z)-\varkappa[p'(z)]^2,$$ where $\varkappa$ is a real number.
We also construct a counterexample to a conjecture by B. Shapiro on the number of real zeroes of the polynomial $F_{\tfrac{n-1}{n}}[p](z)$ in the case when the real polynomial $p$ of degree $n$ has non-real zeroes. We formulate some new conjectures generalising the Hawaii conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_04951 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities Tyaglov, Mikhail Atia, Mohamed J. Classical Analysis and ODEs Primary 12D10, Secondary 26C10, 26C15, 30C15, 30C10 Given a real polynomial $p$ with only real zeroes, we find upper and lower bounds for the number of non-real zeroes of the differential polynomial $$ F_{\varkappa}[p](z):= p(z)p''(z)-\varkappa[p'(z)]^2,$$ where $\varkappa$ is a real number. We also construct a counterexample to a conjecture by B. Shapiro on the number of real zeroes of the polynomial $F_{\tfrac{n-1}{n}}[p](z)$ in the case when the real polynomial $p$ of degree $n$ has non-real zeroes. We formulate some new conjectures generalising the Hawaii conjecture. |
| title | On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities |
| topic | Classical Analysis and ODEs Primary 12D10, Secondary 26C10, 26C15, 30C15, 30C10 |
| url | https://arxiv.org/abs/1912.04951 |