关于H.Weyl和J.Steiner多项式
On H.Weyl and J.Steiner polynomials
摘要：
本文研究两类几何原点的一元多项式的根问题。第一类是Steiner多项式，它由多项式组成，每个多项式都与R^n中的一个紧凸集V相关联。这类多项式将集合V+tB^n的体积描述为t的函数，其中t是正数，B^n表示R中的单位球。第二类是外尔多项式类，由多项式组成，每个多项式与一个黎曼流形M相关联，其中M}等距嵌入R^n的正余维。外尔多项式描述了其相关M的管状邻域的体积作为管的半径的函数。这些多项式计算明确在一些自然的例子，如球，立方体，挤压圆柱体。此外，我们研究了上述多项式如何相互关联，以及它们如何依赖于R^n到R^m的标准嵌入，其中m>n。我们发现，在某些情况下，任何Steiner多项式根的实部将是负的。在某些其他情况下，斯坦纳多项式将只有真正的负根。在所有这些情况下，可以证明外尔多项式的所有根都是单根的，而且它们都位于虚轴上。与此同时，在某些情况下，上述模式并不成立。
英文摘要：
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 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 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 R^n. 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 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.
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