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• W2072338914 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 $$V subset {mathbb{R}}^{n}$$ . 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 $${mathbb{R}}^{n}$$ . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold $${mathcal{M}}$$ , where $${mathcal{M}}$$ is isometrically embedded with positive codimension in $${mathbb{R}}^{n}$$ . A Weyl polynomial describes the volume of a tubular neighborhood of its associated $${mathcal{M}}$$ 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 $${mathbb{R}}^{n}$$ into $${mathbb{R}}^{m}$$ 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. 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." @default.
• W2072338914 created "2016-06-24" @default.
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• W2072338914 date "2008-03-28" @default.
• W2072338914 modified "2023-10-07" @default.
• W2072338914 title "On H. Weyl and J. Steiner Polynomials" @default.
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• W2072338914 doi "https://doi.org/10.1007/s11785-008-0048-z" @default.
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