On multiplier sequences

Abstract

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 ( c 0 , … , c n ) , c i ∈ R , such that if ∑ i = 0 n b i x n − i , b i ∈ R , is a hyperbolic polynomial, then ∑ i = 0 n c i b i x n − i is also such a polynomial. The set of FMS( n + 1 ) coincides with the set of tuples such that ∑ i = 0 n C n i c i x n − i 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).