Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 28, No 2, (2002), 117p-152p
A real polynomial of one real variable is hyperbolic (resp.
strictly hyperbolic) if it has only real roots (resp. if its roots are real and
distinct). We prove that there are 116 possible non-degenerate configurations
between the roots of a degree 5 strictly hyperbolic polynomial and
of its derivatives (i.e. configurations without equalities between roots).
The standard Rolle theorem allows 286 such configurations. To obtain
the result we study the hyperbolicity domain of the family P (x; a, b, c) =
x^5 − x^3 + ax^2 + bx + c (i.e. the set of values of a, b, c ∈ R for which the
polynomial is hyperbolic) and its stratification defined by the discriminant
sets Res(P^(i) , P^(j) ) = 0, 0 ≤ i < j ≤ 4.
∗ Research partially supported by INTAS grant 97-1644