On the Hyperbolicity Domain of the Polynomial x^n + a1x^(n-1) + 1/4+ an

∗ Partially supported by INTAS grant 97-1644

We consider the polynomial Pn = x^n + a1 x^(n−1) + · · · + an , ai ∈ R. We represent by figures the projections on Oa1 . . . ak , k ≤ 6, of its hyperbolicity domain Π = {a ∈ Rn | all roots of Pn are real}. The set Π and its projections Πk in the spaces Oa1 . . . ak , k ≤ n, have the structure of stratified manifolds, the strata being defined by the multiplicity vectors. It is known that for k > 2 every non-empty fibre of the projection Π^k → Π^(k−1) is a segment or a point. We prove that this is also true for the strata of Π of dimension ≥ k. This implies that for any two adjacent strata there always exist a space Oa1 . . . ak , k ≤ n, such that from the projections of the strata in it one is “above” the other w.r.t. the axis Oak . We show 1) how to find this k and which stratum is “above” just by looking at the multiplicity vectors of the strata; 2) how to obtain the relative position of a stratum of dimension l and of all strata of dimension l + 1 and l + 2 to which it is adjacent.