Some remarks about Descartes' rule of signs

TL;DR

This paper refines Descartes' rule of signs to better estimate positive roots of real polynomials and introduces new restrictions on the counts of positive and negative roots, enhancing understanding of polynomial roots.

Contribution

It offers a strengthened, elementary version of Descartes' rule and introduces novel restrictions on the combined positive and negative roots of polynomials.

Findings

A complete, strengthened statement of Descartes' rule for positive roots.

New restrictions on the total number of positive and negative roots.

Enhanced elementary tools for analyzing polynomial roots.

Abstract

What can we deduce about the roots of a real polynomial in one variable by simply considering the signs of its coefficients? On one hand, we give a complete answer concerning the positive roots, by proposing a statement of Descartes' rule of signs which strengthens the available ones while remaining as elementary and concise as the original. On the other hand, we provide new kinds of restrictions on the combined numbers of positive and negative roots.