Counting Zeros of Harmonic Rational Functions and Its Application to Gravitational Lensing

General Relativity gives that finitely many point masses between an observer and a light source create many images of the light source. Positions of these images are solutions of \(r(z)=\bar{z},\) where \(r(z)\) is a rational function. We study the number of solutions to \(p(z) = \bar{z}\) and \(r(z...

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Bibliographic Details
Published in:arXiv.org 2012-12
Main Authors: Bleher, Pavel M, Homma, Youkow, Ji, Lyndon L, Roeder, Roland K W
Format: Article
Language:English
Subjects:
Gravitation
Gravitational lenses
Harmonic functions
Lower bounds
Mathematical analysis
Polynomials
Rational functions
Relativity
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Summary:General Relativity gives that finitely many point masses between an observer and a light source create many images of the light source. Positions of these images are solutions of \(r(z)=\bar{z},\) where \(r(z)\) is a rational function. We study the number of solutions to \(p(z) = \bar{z}\) and \(r(z) = \bar{z},\) where \(p(z)\) and \(r(z)\) are polynomials and rational functions, respectively. Upper and lower bounds were previously obtained by Khavinson-\'{S}wi\c{a}tek, Khavinson-Neumann, and Petters. Between these bounds, we show that any number of simple zeros allowed by the Argument Principle occurs and nothing else occurs, off of a proper real algebraic set. If \(r(z) = \bar{z}\) describes an \(n\)-point gravitational lens, we determine the possible numbers of generic images.
ISSN:2331-8422