On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities

TL;DR

This paper investigates bounds on the number of non-real zeroes of a specific differential polynomial derived from real polynomials with real zeroes, and introduces new conjectures generalizing the Hawaii conjecture.

Contribution

It provides bounds for non-real zeroes of a differential polynomial and constructs a counterexample to a conjecture on real zeroes, extending the theoretical understanding of zero distributions.

Findings

Established bounds for non-real zeroes of the differential polynomial

Constructed a counterexample to Shapiro's conjecture

Formulated new conjectures generalizing the Hawaii conjecture

Abstract

Given a real polynomial $p$ with only real zeroes, we find upper and lower bounds for the number of non-real zeroes of the differential polynomial $$ F_{\varkappa}[p](z):= p(z)p''(z)-\varkappa[p'(z)]^2,$$ where $\varkappa$ is a real number. We also construct a counterexample to a conjecture by B. Shapiro on the number of real zeroes of the polynomial $F_{\tfrac{n-1}{n}}[p](z)$ in the case when the real polynomial $p$ of degree $n$ has non-real zeroes. We formulate some new conjectures generalising the Hawaii conjecture.