On a problem inspired by Descartes' rule of signs

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with \(m\) positive coefficients followed by \(n\) negative followed by \(q\) positive coefficients. We consider the sequence of moduli of their roots on the positive real half-axis; all moduli are supposed distinct. We mark in this sequence the positions of the moduli of the two positive roots. For \(m=n=2\), \(n=q=2\) and \(m=q=2\), we give the exhaustive answer to the question which the positions of the two moduli of positive roots can be.

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Publication Details

Journal
Modern Mathematical Methods
Published
2024-08-14
DOI
https://doi.org/10.64700/mmm.38
Primary Topic
Mathematics and Applications
Type
article
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article

On a problem inspired by Descartes' rule of signs

Vladimir Petrov Kostov
Modern Mathematical Methods
Mathematics and Applications
article

On a problem inspired by Descartes' rule of signs

Vladimir Petrov Kostov
article en

Abstract

We study real univariate polynomials with non-zero coefficients and with all roots real, out of which exactly two positive. The sequence of coefficients of such a polynomial begins with \(m\) positive coefficients followed by \(n\) negative followed by \(q\) positive coefficients. We consider the sequence of moduli of their roots on the positive real half-axis; all moduli are supposed distinct. We mark in this sequence the positions of the moduli of the two positive roots. For \(m=n=2\), \(n=q=2\) and \(m=q=2\), we give the exhaustive answer to the question which the positions of the two moduli of positive roots can be.

Modern Mathematical MethodsVol. 2(2)
Université Côte d'Azur (FR), Observatoire de la Côte d’Azur (FR)
Openalex Percentile: Top 60%
Mathematics and Applications
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