Degree 6 Hyperbolic Polynomials and Orders of Moduli

Yousra Gati; Vladimir Petrov Kostov; Mohamed Chaouki Tarchi

Vol. 29 No. 2 (2024)

Regular Articles

Degree 6 Hyperbolic Polynomials and Orders of Moduli

Published 2024-10-04

Yousra Gati
Vladimir Petrov Kostov
Mohamed Chaouki Tarchi

Yousra Gati
University of Carthage

Vladimir Petrov Kostov
LJAD, CNRAS, Université Côte d'Azur

Mohamed Chaouki Tarchi
LIM-EPT, University of Carthage

Submission of an original manuscript to the Mathematical Communications will be taken to mean that it represents original work not previously published, that it is not being considered elsewhere for publication; and, if accepted for publication, it will be published in print and online form and it will not be published elsewhere in the same form, for commercial purposes, in any language, without the consent of the publisher. The journal takes the stance that the publication of scholarly research is meant to disseminate knowledge and, in a not-for-profit regime, benefits neither publisher nor author financially. It sees itself as having an obligation to its authors and to society to make content available online now that the technology allows for such a possibility.

Abstract

We consider real univariate degree d real-rooted polynomials with non-vanishing coefficients. Descartes’ rule of signs implies that such a polynomial has ˜c positive and ˜p negative roots counted with multiplicity, where ˜c and ˜p are the numbers of sign changes and sign preservations in the sequence of its coefficients, ˜c + ˜p = d. For d = 6, we give the exhaustive answer to the question: When the moduli of all 6 roots are distinct and arranged on the real positive half-axis, in which positions can the moduli of the negative roots be depending on the signs of the coefficients?

Keywords

real polynomial in one variable, hyperbolic polynomial, sign pattern, Descartes’ rule of signs

PDF

Supplementary File(s)

TeX