Descartes' rule of signs, canonical sign patterns and rigid orders of moduli

TL;DR

This paper characterizes when the sign pattern of a real polynomial's coefficients determines the order of its roots' moduli, using a specific forbidden sign pattern criterion.

Contribution

It provides a precise criterion linking coefficient sign patterns to root modulus order for polynomials with distinct positive roots.

Findings

Sign patterns determine root order under specific conditions.

Forbidden sign patterns are identified for root order determination.

The results extend classical Descartes' rule of signs.

Abstract

We consider real polynomials in one variable without vanishing coefficients and with all roots real and of distinct moduli. We show that the signs of the coefficients define the order of the moduli of the roots on the real positive half-line exactly when no four consecutive signs of coefficients equal $(+,+,-,-)$, $(-,-,+,+)$, $(+,-,-,+)$ or $(-,+,+,-)$.