On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture

Olga Katkova; Mikhail Tyaglov; Anna Vishnyakova

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arXiv · 2406.00686

On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture

Olga Katkova·Mikhail Tyaglov·Anna Vishnyakova

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Abstract

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 $H_{\frac{n-1}{n}}[p].$

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Olga Katkova, Mikhail Tyaglov, Anna Vishnyakova. 2024-06-02. On the number of real zeroes of a homogeneous differential polynomial and a generalization of the Hawaii conjecture. https://arxiv.org/abs/2406.00686

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