Spectral methods in the geometry of polynomials

在1960年代，英国的皇家海军B。Sendov解释说，给定任何多项式的次数2或更大的所有其零点都位于单位盘上的复平面，每个根的多项式是一个距离最多为一个零的衍生物的多项式。森多夫猜想仍然没有解决（虽然它已经被证明是次数小于或等于8的多项式），被认为是复变函数理论中的一个重要问题。在过去的两年中，研究人员已经设计出了使用矩阵来研究多项式的零点与其导数的零点之间的关系的方法。我建议开发技术矩阵分析，特别是在谱扰动理论，这可能是有用的解决森多夫猜想。虽然这主要是好奇心驱动的研究，光谱微扰理论在物理科学和计算中有重要的应用，开发的技术可能会在这些领域中找到应用。

In the 1960's the mathematican B. Sendov conjectured that given any polynomial of degree two or greater all of whose zeros lie in the unit disk on the complex plane, each root of the polynomial is a distance of at most one from a zero of the derivative of that polynomial. Sendov's conjecture remains unsolved (although it has been proven for polynomials of degree less than or equal to eight) and is considered an important problem in complex function theory. In the past two years, researchers have devised ways of using matrices to study the relationship between the zeros of a polynomial and that of its derivative. I propose to develop techniques in matrix analysis, especially in spectral perturbation theory, that may be useful in solving Sendov's conjecture. While this is primarily curiosity-driven research, spectral perturbation theory has important applications in the physical sciences and computation and the techniques developed may find application in these areas.