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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
DOI: https://doi.org/10.1090/proc/15215
Published electronically: September 18, 2020
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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
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

DOI: https://doi.org/10.1090/proc/15215
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