Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 25, No 1, (1999), 47p-70p
Abstract:
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.
