  We study a natural stratification of certain affine slices of univariate
hyperbolic polynomials. We look into which posets of strata can be realized and
show that the dual of the poset of strata is a shellable simplicial complex and
in particular a combinatorial sphere. From this we obtain a g-theorem and an
upper bound theorem on the number of strata. We use these results to design
smaller test sets to improve upon Timofte's degree principle and give bounds on
how much the degree principle can be improved.
