Critical Points, Critical Values, and a Determinant Identity for Complex Polynomials

TL;DR

This paper extends the understanding of the map from critical points to critical values of complex polynomials, including cases with multiplicities, and connects the determinant identity to Dyson's conjecture.

Contribution

It generalizes the local homeomorphism property and determinant identity for polynomials with multiple critical points, broadening previous results.

Findings

Extended the local homeomorphism result to polynomials with multiplicities.

Derived a generalized determinant identity related to Dyson's conjecture.

Connected the polynomial critical point map to stratification based on critical point multiplicities.

Abstract

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 $\theta\colon \mathbb{C}^n\to \mathbb{C}^n$ 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.