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

Abstract

For a given real polynomial $p$ we study the possible number of real roots of a differential polynomial $H_{\varkappa }[p](x)=\varkappa {\left(p^{\prime }(x)\right)}^2-p(x)p^{\prime \prime }(x),\varkappa \in \mathbb{R}.$ In the special case when all real zeros of the polynomial $p$ are simple, and all roots of its derivative $p^{\prime }$ 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 $H_{\frac{n-1}{n}}[p].$