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

For a given real polynomial $p$ we study the possible number of real roots of a differential polynomial $H_{\mathrm{\varkappa }}[p](x)=\mathrm{\varkappa }{(p^{\mathrm{\prime }}(x))}^2-p(x)p^{\mathrm{\prime }\mathrm{\prime }}(x),\mathrm{\varkappa }\in \mathbb{R}\mathrm{.}$ In the special case when all real zeros of the polynomial $p$ are simple, and all roots of its derivative $p^{\mathrm{\prime }}$ are real and simple, the distribution of zeros of $H_{\mathrm{\varkappa }}[p]$ is completely described for each real $\mathrm{\varkappa }\mathrm{.}$ We also provide counterexamples to two Boris Shapiro’s conjectures about the number of zeros of the function $H_{\frac{n-1}{n}}[p]\mathrm{.}$