Hawaii conjecture through the lens of Cauchy indices

TL;DR

This paper investigates the properties of real critical points of the logarithmic derivative of a polynomial using Cauchy indices, leading to improved bounds on their number.

Contribution

It introduces a novel application of Cauchy indices to analyze critical points of polynomial derivatives, enhancing existing bounds.

Findings

Improved lower bound for the number of critical points of the logarithmic derivative.

Application of Cauchy indices to polynomial critical point analysis.

Deeper understanding of the distribution of critical points in real polynomials.

Abstract

Given a real polynomial $p$, we study some properties of real critical points of its logarithmic derivative $Q[p]=(p'/p)'$ using the theory of Cauchy indices. As a by-product we improve the lower bound for the number these points.