Complex Analysis and Operator Theory

Abstract

The paper deals with root location problems for two classes of uni- variate 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 ⊂ R n . A polynomial of this class describes the volume of the set V + tB n as a function of t ,w heret is a positive number and B n denotes the unit ball in R n . The second class, the class of Weyl polynomi- als, consists of polynomials, each associated with a Riemannian manifold M, where M is isometrically embedded with positive codimension in R n .AW eyl 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. Fur- thermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of R n into R m 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.