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，复平面上每一个包含多项式p(z)的零点的圆域，都包含它的任何一个极多项式的零点。最近，我们引入了复数多项式p(z)轨迹的概念。它是复平面上最小的闭集（相对于包含），它包含它的任何一个极多项式的零。我们建立了轨迹的几个一般性质，并特别证明了集合是多项式轨迹的性质在莫比乌斯变换下是守恒的，每个轨迹都是其内部的闭包，每个轨迹都是在轨迹外具有极点的p(z)的所有极导数的零的闭包。我们还展示了轨迹的概念与多项式几何上其他几个经典定理之间的联系，如拉盖尔定理、罗尔定理、格雷斯-塞戈-沃尔什重合定理。在每一个例子中，轨迹的概念都提供了一个最小集合，其中每一个定理都成立。多项式的所有轨迹的类是非常丰富的，具有有趣的性质。我们的第一个目标是阐明这些特性，分离和研究几个基因座的子类，如最小面积的基因座、光滑边界的基因座或对称的基因座。另一个目标是为多项式轨迹的（近似）计算开发计算效率高的算法，也称为轨迹保持器。轨迹和多项式的极坐标导数的零点之间的联系提出了一种攻击1962年Bl. Sendov猜想的方法。

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.