Some Generalizations of the Hawaii Conjecture and BeyondSome Generalizations of the Hawaii Conjecture and BeyondO. Katkova et al.

For a given real polynomial p we study the possible number of real roots of a differential polynomial Hϰ[p](x)=ϰp′(x)2-p(x)p′′(x),ϰ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\varkappa }[p](x) = \varkappa \left( p'(x)\right) ^2-p(x)p''(x), \varkappa \in \mathbb {R}$$\end{document}. In the special case when all real zeros of the polynomial p are simple, and all roots of its derivative p′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p'$$\end{document} are real and simple, the distribution of zeros of Hϰ[p]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\varkappa }[p]$$\end{document} is completely described for each real ϰ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varkappa $$\end{document}. We also provide counterexamples to two Boris Shapiro’s conjectures about the number of zeros of the function Hn-1n[p]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\frac{n-1}{n}}[p]$$\end{document}.