# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Critical points, critical values, and a determinant identity for complex polynomialsHTML articles powered by AMS MathViewer

by Michael Dougherty and Jon McCammond
Proc. Amer. Math. Soc. 148 (2020), 5277-5289 Request permission

## Abstract:

Given any $n$-tuple of complex numbers, one can easily define a canonical 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.
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• Michael Dougherty
• Affiliation: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081
• MR Author ID: 938590
• Email: mdoughe1@swarthmore.edu
• Jon McCammond
• Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
• MR Author ID: 311045
• Email: jon.mccammond@math.ucsb.edu
• Received by editor(s): September 5, 2019
• Received by editor(s) in revised form: June 2, 2020
• Published electronically: September 18, 2020
• Communicated by: Harold P. Boas