Sign patterns and rigid moduli orders

TL;DR

This paper studies the geometric structure of the hyperbolicity domain of real polynomials and the subset where roots have equal moduli, revealing local intersection properties and singularities.

Contribution

It characterizes the local geometric and singularity structure of the set of polynomials with roots of equal moduli, including intersection and Whitney umbrella singularities.

Findings

Set $E_d$ is locally the intersection of smooth hypersurfaces at points with multiple root equalities.

At points with two double roots, $E_d$ has a Whitney umbrella singularity.

Visualizations provided for degrees up to 4.

Abstract

We consider the set of monic degree $d$ real univariate polynomials $Q_d=x^d+\sum_{j=0}^{d-1}a_jx^j$ and its {\em hyperbolicity domain} $\Pi_d$, i.e. the subset 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 draw pictures of the sets $\Pi_d$ and~$E_d$.