Descartes' rule of signs is exact!
type: Item
instance of: scholarly article
title: Descartes' rule of signs is exact!
zbMATH Open document ID: 1214.26010
published in: Journal of Algebra
publication date: 2011-01-21T00:00:00Z
review text: Descartes' rule of signs states that for any non-zero polynomial \(f\in\mathbb{R}[x]\) with \(t\) positive roots counted with multiplicities, the difference between the number of sign changes in the vector of coefficients of \(f\), denoted by \(S(f)\), and \(t\) is a non-negative even integer. Poincaré showed that this rule can be made exact, i.e., when the two quantities match up completely, by premultiplying \(f\) by a correction factor that depends on \(f\). The author first sharpens Poincaré's result and shows that the correction factor can be chosen as a polynomial with non-negative coefficients. As a second main result, the following general statement is proven: For any fixed \(f\in\mathbb{R}[x]\), the sequence \(S((x+1)^n f(x))\), \(n=0,1,\dots\) is monotone decreasing with limit equal to the number of positive roots of \(f\) counted with multiplicities. The paper is very well written and has a nice historical introduction to the topic.
reviewed by: Thomas Stoll
Mathematics Subject Classification ID: 26C10
zbMATH DE Number: 5840425
zbMATH Keywords: Descartes' rule of signs
zbMATH Keywords: root counting
author: Martín Avendaño
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.08.014
OpenAlex ID: W2020406245
cites work: An Inductive Proof of Descartes' Rule of Signs
cites work: Descartes' Rule of Signs Revisited
cites work: The number of roots of a lacunary bivariate polynomial on a line
cites work: A quantitative Pólya's theorem with zeros
cites work: Descartes' Rule of Signs: Another Construction
cites work: A new bound for Pólya's theorem with applications to polynomials positive on polyhedra.
cites work: A Simple Proof of Descartes's Rule of Signs
DOI: 10.1016/J.JALGEBRA.2010.08.014
description: scientific article; zbMATH DE number 5840425