On a problem inspired by Descartes' rule of signs

TL;DR

This paper investigates the arrangement of roots of real polynomials with specific sign patterns and exactly two positive roots, providing exhaustive classifications for certain coefficient configurations.

Contribution

It offers a detailed analysis and classification of the possible positions of positive roots' moduli in polynomials with prescribed sign patterns and root counts.

Findings

Classifies root modulus arrangements for specific coefficient sign patterns.

Provides exhaustive answers for particular cases of coefficient sequences.

Enhances understanding of root distribution in sign-constrained polynomials.

Abstract

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with $m$ positive coefficients followed by $n$ negative followed by $q$ positive coefficients. We consider the sequence of moduli of their roots on the positive real half-axis; all moduli are supposed distinct. We mark in this sequence the positions of the moduli of the two positive roots. For $m=n=2$, $n=q=2$ and $m=q=2$, we give the exhaustive answer to the question which the positions of the two moduli of positive roots can be.