# Descartes' rule of signs and moduli of roots

## TL;DR

This paper investigates the relationship between the sign patterns of coefficients in hyperbolic polynomials and the possible arrangements of their roots' moduli, focusing on cases with one or two sign changes.

## Contribution

It provides a detailed analysis of how the sign change pattern constrains the moduli positions of roots in hyperbolic polynomials with specific sign change counts.

## Key findings

- Characterization of root moduli positions for c=1 and c=2 cases
- Conditions linking coefficient signs to root arrangements
- Insights into root distribution based on sign patterns

## Abstract

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. For $c=1$ and $2$, we discuss the question: When the moduli of all the roots of a HP are arranged in the increasing order on the real half-line, at which positions can be the moduli of its positive roots depending on the positions of the sign changes in the sequence of coefficients?