On H. Weyl and J. Steiner Polynomials |
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Authors: | Victor Katsnelson |
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Institution: | (1) Department of Mathematics, The Weizmann Institute, Rehovot, 76100, Israel |
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Abstract: | The paper deals with root location 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 . A polynomial of this class describes the volume of the set V + tB
n
as a function of t, where t is a positive number and B
n
denotes the unit ball in . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold , where is isometrically embedded with positive codimension in . A Weyl polynomial describes the volume of a tubular neighborhood of its associated 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 into 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.
Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual
experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole
day. The experiment yielded no results.
Submitted: March 5, 2007., Revised: February 1, 2008., Accepted: February 2, 2008. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 53C99 52A39 30C10 Secondary 52A22 60D05 |
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