Sign patterns and rigid moduli orders

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~\(E_d\).