Hyperbolic polynomials and starved polytopes

2023/07/06 by Arne Lien, Lien, Arne
Computer Science · Mathematics · #05E14 #14P10 #20C30 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric and Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2307.03239

openalex publication_date 2023/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We study sets of univariate hyperbolic polynomials that share the same first few coefficients and show that they have a natural combinatorial description akin to that of polytopes. We define a stratification of such sets in terms of root arrangements of hyperbolic polynomials and show that any stratum is either empty, a point or of maximal dimension and in the latter case we characterise its relative interior. This is used to show that the poset of strata is a graded, atomic and coatomic lattice and to provide an algorithm for computing which root arrangements are realised in such sets of hyperbolic polynomials.

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