On multiplier sequences

In the present paper we consider real polynomials in one real variable of a given degree n. Such a polynomial is called hyperbolic if it has only real roots. A finite multiplier sequence of length n+1 (FMS(n+1)) is a tuple (c0,…,cn), , such that if , , is a hyperbolic polynomial, then is also such a polynomial. The set of FMS(n+1) coincides with the set of tuples such that is a hyperbolic polynomial with all roots of the same sign. In the paper we prove several geometric properties of the set of FMS(n+1) formulated in terms of its stratification (defined by the multiplicity vectors of the polynomials) and of the Whitney property (the curvilinear distance to be equivalent to the Euclidean one).