Critical points, critical values, and a determinant identity for complex polynomials,Proceedings of the American Mathematical Society

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Critical points, critical values, and a determinant identity for complex polynomials
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-09-18 , DOI: 10.1090/proc/15215
Michael Dougherty , Jon McCammond

Abstract:Given any

-tuple of complex numbers, one can easily define a canonical polynomial of degree

that has the entries of this

-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta \colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $\theta$ . In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb{C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb{C}^\ell \to \mathbb{C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture.

中文翻译：

复杂多项式的临界点，临界值和行列式标识

摘要：给定任何

复数元组，可以轻松定义以该元组

的条目

作为临界点的次数的规范多项式。2002年，Beardon，Carne和Ng研究了一个映射，该映射输出根据输入构造的规范多项式的临界值，并证明该映射已存在。一路走来，他们表明只要输入的输入项不同且非零，它就是局部同胚，并且隐式地为的雅可比行列式生成了多项式表达式。在本文中，我们将局部同胚性结果和优雅的行列式同一性扩展并推广到相似的情况，在这些情况下，临界点以多重性出现。这涉及分层 $\ theta \冒号\ mathbb {C} ^ n \ to \ mathbb {C} ^ n$ $\ theta$ $\ theta$ $\ mathbb {C} ^ n$ 根据哪个坐标相等，并概括 $\ theta$ 到一个相似的图，其中是不同的关键点的数量。我们建立的更复杂的行列式恒等式与被称为戴森猜想的多项式恒等式紧密相关。 $\ mathbb {C} ^ \ ell \ to \ mathbb {C} ^ \ ell$ $\ ell$

更新日期：2020-10-19

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