Loci of Complex Polynomials, Generalized Primary Matrix Functions, and Connections with Bernstein Functions and Levy Processes

经典的格雷斯定理指出，复平面中的每个圆形区域包含多项式p（z）的零点，包含其任何非极多项式的零点。最近，我们引入了复多项式p（z）的轨迹的概念。它是复平面中包含任何非极多项式的零点的最小（相对于包含）闭集。我们建立了几个一般性质的轨迹，并表明，特别是，财产的一套是一个轨迹的多项式是保持下的莫比乌斯变换，每一个轨迹是封闭的内部，每一个轨迹是封闭的零点的所有极导数的p（z）有极点以外的轨迹。本文还指出了轨迹概念与多项式几何中的几个经典定理，如Laguerre定理、Rolle定理、Grace-Szego-沃尔什重合定理之间的联系。在每一种情况下，轨迹的概念都提供了一个最小集合，这些定理中的每一个都对它成立。多项式的所有轨迹的类是一个非常丰富的具有有趣的性质。我们的第一个目标是阐明这些性质，分离和研究几个子类的轨迹，如轨迹与最小面积，轨迹与光滑边界，或轨迹的对称性。另一个目标是开发用于多项式的轨迹（也称为轨迹保持器保持器）的（近似）计算的计算高效算法。多项式的极导数的零点和轨迹之间的连接表明了攻击BI的方法。1962年的森多夫猜想。

The classical theorem of Grace states that every circular domain in the complex plane containing the zeros of a polynomial p(z), contains a zero of any of its apolar polynomials. Recently, we introduced the notion of a locus of a complex polynomial p(z). It is a smallest (with respect to inclusion) closed set in the complex plane that contains a zero of any of its apolar polynomials. We established several general properties of the loci and showed, in particular, that the property of a set being a locus of a polynomial is preserved under a Mobius transformation, that every locus is the closure of its interior, and that every locus is the closure of the zeros of all polar derivatives of p(z) having poles outside of the locus. We also showed the connection between the notion of the locus and several other classical theorems on Geometry of Polynomials, such as Laguerre's theorem, Rolle's theorem, Grace-Szego-Walsh Coincidence theorem. In every instance the notion of a locus provides a minimal set for which each one of these theorems holds. The class of all loci of a polynomial is a very rich having intriguing properties. Our first goal is to shed light on these properties, isolate and investigate several subclasses of loci, such as the locus with the smallest area, loci with smooth boundary, or loci with symmetries. Another goal is to develop computationally efficient algorithms for the (approximate) computation of a locus of a polynomial, also known as a locus holder. The connections between a locus and the zeros of the polar derivatives of a polynomial suggests an approach for attacking the Bl. Sendov's Conjecture dating back to 1962.