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

TL;DR

This paper investigates the number of real roots of a differential polynomial derived from a real polynomial, providing a complete description under certain conditions and counterexamples to existing conjectures.

Contribution

It offers a complete characterization of zeros distribution for a class of differential polynomials and disproves two conjectures by Boris Shapiro.

Findings

Distribution of zeros fully described for simple roots case

Counterexamples to Shapiro's conjectures provided

Conditions for real roots of differential polynomials established

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