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 (c(0),center dot center dot center dot, c(n)), c(i) is an element of R, such that if Sigma(n)(i=0) b(i)x(n-i),b(i) is an element of R, is a hyperbolic polynomial, then Sigma(n)(i=0) 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 Sigma(n)(i=0) 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). (c) 2007 Elsevier B.V. All rights reserved.
finite multiplier sequence hyperbolic polynomial stratification multiplicity vector hyperbolicity domain