  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.
