On H.Weyl and J.Steiner polynomials
Abstract
The paper deals with root problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set V in Rn. A polynomial of this class describes the volume of the set V+tBn as a function of t, where t is a positive number and Bn denotes the unit ball in R. The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold M, where M is isometrically embedded with positive codimension in Rn. A Weyl polynomial describes the volume of a tubular neighborhood of its associated M as a function of the tube's radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of Rn into Rm for m>n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial's roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.