Hyperbolic polynomials and starved polytopes

TL;DR

This paper explores the structure of sets of hyperbolic polynomials sharing initial coefficients, revealing a combinatorial and lattice structure that aids in understanding root arrangements and their realizability.

Contribution

It introduces a stratification of hyperbolic polynomial sets based on root arrangements and characterizes their geometric and combinatorial properties.

Findings

Strata are either empty, points, or of maximal dimension.

The poset of strata forms a graded, atomic, and coatomic lattice.

An algorithm is provided to determine realizable root arrangements.

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.