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

经典的Grace定理指出，复平面中包含多项式p(Z)的零点的每个圆域都包含它的任一非极多项式的零点。最近，我们引入了复多项式p(Z)的轨迹的概念。它是复平面中最小的(关于包含)闭集，它包含它的任一非极多项式的零。我们建立了轨迹的几个一般性质，特别地，证明了集合是多项式的轨迹的性质在Mobius变换下保持不变，每个轨迹都是其内部的闭包，并且每个轨迹都是p(Z)的所有极导数的零点的闭包。给出了轨迹概念与拉盖尔定理、罗尔定理、Grace-Szego-Walsh重合定理等几个多项式几何经典定理之间的联系。在每一种情况下，轨迹的概念都提供了一个极小集，其中每个定理都成立。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性质，分离和研究轨迹的几个子类，如面积最小的轨迹、边界光滑的轨迹或对称的轨迹。另一个目标是开发计算高效的算法，用于(近似)计算多项式的轨迹，也称为轨迹保持器。多项式的极导数的轨迹和零点之间的联系给出了一种攻击B1的方法。Sendov的猜想可以追溯到1962年。*我正在追求的另一个研究方向是关于谱函数的性质(SPF)。SPF是关于厄米矩阵变元的函数，在么正相似变换下不变。每个SPF可以用唯一的方式表示为R^n上的对称函数与厄米特矩阵的特征值的组合。这些函数在经典的复分析、最优化、非光滑和矩阵分析、弹性力学、统计学和量子物理等领域有着广泛的应用，特别是如果R^n上的对称函数是可分的，那么我们就得到了所谓的可分SPF。通过它们的导数，后者与在算子单调理论和算子凸函数理论中得到应用的初等矩阵函数联系起来。SPF和主矩阵函数之间的已知连接不能令人满意。我目前正在研究一类矩阵值映射族，它推广了SPF和主矩阵函数。这使得SPF和初等矩阵函数的整个理论得以扩展，特别是导致了算子单调和算子凸函数的经典概念的推广。

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.******Another research direction that I am pursuing concerns the properties of spectral functions (SpF). A SpF is a function on a Hermitian matrix argument, invariant under unitary similarity transformation. Each SpF can be represented in a unique way as the composition of a symmetric function on R^n with the eigenvalues of a Hermitian matrices. These functions find numerous applications in diverse areas such as classical complex analysis, optimization, non-smooth and matrix analysis, elasticity, statistics, and quantum physics.  If, in particular, the symmetric function on R^n is separable, then we obtain the so-called separable SpF. Through their derivative, the latter are connected with the primary matrix functions finding applications in the theory of operator monotone and operator convex functions. The known connection between the SpF's and the primary matrix functions is unsatisfactory. I am currently working on a family of matrix-valued maps that generalizes the SpF's and the primary matrix functions. This allows the entire theory of SpF's and the primary matrix functions to be extended, and in particular leads to a generalization of the classical notions of operator monotone and operator convex functions.**