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Ji, Pavel M. Bleher, Roland K. W. Roeder, Youkow Homma","submitted_at":"2012-06-11T16:10:29Z","abstract_excerpt":"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. 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