On the geometric properties of Vandermonde's mapping and on the problem of moments

Abstract

Synopsis In this paper we prove that the domain of hyperbolicity of the polynomial x n + λ 2 n n−2 +λ 3 x n−3 + … + λ n ,λ i ϵ R intersected by the half-space λ 2 ≧ – 1, has the property of Whitney, i.e., every two points of this set can be connected by a piecewise-smooth curve belonging to it, whose length is ≦ C times greater than the euclidian distance between the points, where the constant C does not depend on the choice of the points. Parallel with this, we show that the values x 1 ≦x 2 ≦…≦x n of a random variable are uniquely determined by the corresponding probabilities and by thefirst n moments.