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Critical points, critical values, and a determinant identity for complex polynomials

Authors: Michael Dougherty and Jon McCammond
Journal: Proc. Amer. Math. Soc. 148 (2020), 5277-5289
MSC (2010): Primary 30C10, 30C15; Secondary 05A10, 57N80
Published electronically: September 18, 2020
MathSciNet review: 4163840
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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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Additional Information

Michael Dougherty
Affiliation: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081
MR Author ID: 938590

Jon McCammond
Affiliation: Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106
MR Author ID: 311045

Keywords: Polynomials, critical values, Dyson’s conjecture, stratifications
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
Article copyright: © Copyright 2020 American Mathematical Society