On H.Weyl and J.Steiner polynomials - Mathematics > Classical Analysis and ODEsReport as inadecuate




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Abstract: The paper deals with root problems for two classes of univariate polynomialsboth of geometric origin.The first class discussed, the class of Steiner polynomial, consists ofpolynomials, each associated with a compact convex set V in R^n. A polynomialof this class describes the volume of the set V+tB^n as a function of t, wheret is a positive number and B^n denotes the unit ball in R. The second class,the class of Weyl polynomials, consists of polynomials, each associated with aRiemannian manifold M, where M} is isometrically embedded with positivecodimension in R^n. A Weyl polynomial describes the volume of a tubularneighborhood of its associated M as a function of the tube-s radius. Thesepolynomials are calculated explicitly in a number of natural examples such asballs, cubes, squeezed cylinders. Furthermore, we examine how the abovementioned polynomials are related to one another and how they depend on thestandard embedding of R^n into R^m for m>n. We find that in some cases the realpart of any Steiner polynomial root will be negative. In certain other cases, aSteiner polynomial will have only real negative roots. In all of this cases, itcan 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 theabove pattern does not hold.



Author: Victor Katsnelson

Source: https://arxiv.org/



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