On arrangements of the roots of a hyperbolic polynomial and of one of its derivatives

2002/11/07 by Vladimir Petrov Kostov, Kostov, Vladimir Petrov
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical Dynamics and Fractals #Polynomial and algebraic computation #math.AG

paper · pdf · doi:10.48550/arxiv.math/0211132

arxiv created 2002/11/07 · openalex publication_date 2002/11/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider real monic \em hyperbolic polynomials in one real variable, i.e. polynomials having only real roots. Call \em hyperbolicity domain Π of the family of polynomials P(x,a)=xⁿ+a₁x^n-1+... +a_n, a_i,x∈ \bf R, the set \a∈ \bf Rⁿ| P is hyperbolic \. The paper studies a stratification of Π defined by the arrangement of the roots of P and P^(k), where 2≤ k≤ n-1. We prove that the strata are smooth contractible real algebraic varieties.

Related

• Hyperbolic polynomials and starved polytopes
• Combinatorics and topology of stratifications of the space of monic polynomials with real coefficients
• Topology of spaces of hyperbolic polynomials and combinatorics of resonances
• On a stratification defined by real roots of polynomials
• On arrangements of real roots of a real polynomial and its derivatives