Sign Patterns and Rigid Orders of Moduli

Sign Patterns and Rigid Orders of Moduli

Yousra Gati, Vladimir Petrov Kostov, Mohamed Chaouki Tarchi – GJM, Volume 6, Issue 1 (2021), 60-72.

We consider the set of monic degree d real univariate polynomials Q_d=x^d+\sum _{j=0}^{d-1}a_jx^j and their hyperbolicity domains \Pi _d , i.e. the subsets of values of the coefficients a_j for which the polynomial Q_d has all roots real. The subset E_d\subset \Pi _d is the one on which a modulus of a negative root of Q_d is equal to a positive root of Q_d . At a point, where Q_d has d distinct roots with exactly s ( 1\leq s\leq [d/2] ) equalities between positive roots and moduli of negative roots, the set E_d is locally the transversal intersection of s smooth hypersurfaces. At a point, where Q_d has two double opposite roots and no other equalities between moduli of roots, the set E_d is locally the direct product of \mathbb{R}^{d-3} and a hypersurface in \mathbb{R}^3 having a Whitney umbrella singularity. For d\leq 4 , we describe the hyperbolicity domains in terms of sign patterns and (generalized) orders of moduli, and we draw pictures of the sets \Pi _d and E_d .

Categories: Issue1

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Received: March 10, 2021
Accepted: July 24, 2021
Published online: September 8, 2021

Authors:

Yousra Gati
Universite de Carthage, EPT-LIM, Tunisie
Vladimir Petrov Kostov
Université Côte d'Azur, CNRS, LJAD, France
Mohamed Chaouki Tarchi
Université de Carthage, EPT-LIM, Tunisie.

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