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2018 A generalization of the Bocher-Grace theorem
John Clifford, Michael Lachance
Rocky Mountain J. Math. 48(4): 1069-1076 (2018). DOI: 10.1216/RMJ-2018-48-4-1069

Abstract

The Bocher-Grace theorem can be stated as follows: let $p$ be a third degree complex polynomial. Then, there is a unique inscribed ellipse interpolating the midpoints of the triangle formed from the roots of $p$, and the foci of the ellipse are the critical points of $p$. Here, we prove the following generalization: let $p$ be an $n$th degree complex polynomial, and let its critical points take the form \[ \alpha +\beta \cos k\pi /n,\quad k=1,\ldots ,n-1,\ \beta \ne 0. \] Then, there is an inscribed ellipse interpolating the midpoints of the convex polygon formed by the roots of $p$, and the foci of this ellipse are the two most extreme critical points of $p$: $\alpha \pm \beta \cos \pi /n$.

Citation

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John Clifford. Michael Lachance. "A generalization of the Bocher-Grace theorem." Rocky Mountain J. Math. 48 (4) 1069 - 1076, 2018. https://doi.org/10.1216/RMJ-2018-48-4-1069

Information

Published: 2018
First available in Project Euclid: 30 September 2018

zbMATH: 06958769
MathSciNet: MR3859748
Digital Object Identifier: 10.1216/RMJ-2018-48-4-1069

Subjects:
Primary: 51E10

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 4 • 2018
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