复多项式的临界点、临界值和一个行列式恒等式
Critical Points, Critical Values, and a Determinant Identity for Complex Polynomials
摘要：
给定任何n元组的复数，可以规范地定义一个n+1次多项式，该多项式以该n元组的元素作为其临界点。2002年，Beardon，Carne和Ng研究了一个输出由输入构造的标准多项式的临界值的映射$θ\colon \mathbb{C}^n\to \mathbb{C}^n$，并证明了这个映射是上的。在此过程中，他们证明了只要输入的元素是不同的且非零的，$θ$就是一个局部同胚，并且，他们隐式地为$θ$的雅可比行列式生成了一个多项式表达式。本文将局部同胚结果和优美行列式恒等式推广到临界点重数出现的类似情形。这涉及根据坐标相等对$\mathbb{C}^n$进行分层，并将$θ$推广到类似的地图$\mathbb{C}^\ell \to \mathbb{C}^\ell$，其中$\ell$是不同临界点的数量。我们建立的更复杂的行列式恒等式与称为戴森猜想的多项式恒等式密切相关。
英文摘要：
Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $θ\colon \mathbb{C}^n\to \mathbb{C}^n$ 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 $θ$ 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 $θ$. 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 $θ$ 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.
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