Spectral methods in the geometry of polynomials

在20世纪60年代，数学家B. Sendov推测，给定任何二阶或以上的多项式，其所有的零都在复平面上的单位圆盘上，该多项式的每个根与该多项式的导数的零的距离最多为1。Sendov猜想至今仍未得到解决（尽管它已被证明适用于小于或等于8次的多项式），并且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，研究人员发明了使用矩阵来研究多项式的零与其导数的零之间关系的方法。我建议在矩阵分析中发展技术，特别是在谱摄动理论中，这可能对解决森多夫猜想有用。虽然这主要是好奇心驱动的研究，谱摄动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.