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

复多项式的轨迹、广义初等矩阵函数以及与 Bernstein 函数和 Levy 过程的联系

• 批准号: RGPIN-2015-04540
• 负责人: Sendov, Hristo
• 金额: $ 1.02万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635223
• 关键词: Loci; Complex; Polynomials; Generalized; Primary

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.

经典格雷斯定理指出，复平面中包含多项式 p(z) 的零点的每个圆形域都包含其任何非极性多项式的零点。最近，我们引入了复多项式 p(z) 的轨迹的概念。它是复平面中最小的（相对于包含而言）闭集，其中包含任何非极性多项式的零。我们建立了轨迹的几个一般性质，并特别表明，集合作为多项式轨迹的性质在莫比乌斯变换下得以保留，每个轨迹都是其内部的闭包，并且每个轨迹都是极点位于该轨迹之外的 p(z) 的所有极导数的零点的闭包。我们还展示了轨迹概念与多项式几何中其他几个经典定理之间的联系，例如拉盖尔定理、罗尔定理、格蕾丝-塞戈-沃尔什符合定理。在每种情况下，轨迹的概念都提供了每个定理都成立的最小集合。多项式的所有轨迹的类别非常丰富，具有有趣的性质。我们的首要目标是阐明这些特性，分离并研究基因座的几个子类，例如面积最小的基因座、具有平滑边界的基因座或具有对称性的基因座。另一个目标是开发计算高效的算法，用于多项式轨迹（也称为轨迹保持器）的（近似）计算。多项式极导数的轨迹和零点之间的联系提出了一种攻击 BL 的方法。森多夫猜想可以追溯到 1962 年。