Beyond Descartes' rule of signs

TL;DR

This paper investigates the relationship between the sign patterns of coefficients of real univariate polynomials with all real roots and the possible positions of negative roots, providing new insights into root distribution constraints.

Contribution

It introduces novel results on how the known sign change positions influence the moduli of negative roots in such polynomials.

Findings

New bounds on negative root moduli based on coefficient sign changes

Characterization of root position constraints for polynomials with all real roots

Extension of classical Descartes' rule of signs to more detailed root distribution insights

Abstract

We consider real univariate polynomials with all roots real. Such a polynomial with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients has $c$ positive and $p$ negative roots counted with multiplicity. Suppose that all moduli of roots are distinct; we consider them as ordered on the positive half-axis. We ask the question: If the positions of the sign changes are known, what can the positions of the moduli of negative roots be? We prove several new results which show how far from trivial the answer to this question is.