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Conformal Geometry and Dynamics

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The location of critical points of finite Blaschke products

Author: David A. Singer
Journal: Conform. Geom. Dyn. 10 (2006), 117-124
MSC (2000): Primary 53A35; Secondary 30D50
Published electronically: June 7, 2006
MathSciNet review: 2223044
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Abstract | References | Similar Articles | Additional Information

Abstract: A theorem of Bôcher and Grace states that the critical points of a cubic polynomial are the foci of an ellipse tangent to the sides of the triangle joining the zeros. A more general result of Siebert and others states that the critical points of a polynomial of degree $ N$ are the algebraic foci of a curve of class $ N-1$ which is tangent to the lines joining pairs of zeroes. We prove the analogous results for hyperbolic polynomials, that is, for Blaschke products with $ N$ roots in the unit disc.

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Additional Information

David A. Singer
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058

Received by editor(s): January 16, 2006
Published electronically: June 7, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.