# 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, 2021