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Vol. 46 No. 2 (2020)
Serdica Mathematical Journal

Articles

Hyperbolic polynomials and canonical sign patterns

Published 18-09-2020

  • Vladimir Petrov Kostov

Vladimir Petrov Kostov
Université Côte d’Azur, France

Abstract

A real univariate polynomial is hyperbolic if all its roots are real. By Descartes' rule of signs a hyperbolic polynomial (HP) with all coefficients nonvanishing has exactly \(c\) positive and exactly \(p\) negative roots counted with multiplicity, where \(c\) and \(p\) are the numbers of sign changes and sign preservations in the sequence of its coefficients. We discuss the question: If the moduli of all \(c+p\) roots are distinct and ordered on the positive half-axis, then at which positions can the \(p\) moduli of negative roots be depending on the positions of the positive and negative signs of the coefficients of the polynomial? We are especially interested in the choices of these signs for which exactly one order of the moduli of the roots is possible.

2020 Mathematics Subject Classification:

26C10, 30C15

Keywords

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

Full text

Author Details

Vladimir Petrov Kostov

Université Côte d’Azur
Laboratoire de Mathematiques “Jean-Alexandre Dieudonne”
06108 Nice, France
email: vladimir.kostov@unice.fr

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