Spectral methods in the geometry of polynomials

20世纪60年代，数学家B·森多夫在S提出猜想，给定任意二次及以上的多项式，其零点全部位于复平面上的单位圆盘上，则该多项式的每个根至多距该多项式的导数的零点至多一距离。森多夫猜想仍未解决(尽管它已被证明适用于次数小于或等于8的多项式)，并被认为是复函数论中的一个重要问题。在过去的两年里，研究人员已经设计出了使用矩阵来研究多项式的零点和它的导数的零点之间的关系的方法。我建议发展矩阵分析的技术，特别是谱微扰理论，这可能有助于解决森多夫猜想。虽然这主要是好奇心驱动的研究，但光谱微扰理论在物理科学和计算中有重要的应用，所开发的技术可能会在这些领域找到应用。

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.