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

General Relativity gives that finitely many point masses be tween an observer and a light source create many images of the light source. Positions of these images are solutions of r(z) = ¯ where r(z) is a rational function. We study the number of solutions to p(z) = ¯ z and r(z) = ¯ where p(z) and r(z) are polynomials and rational functions, respectively. Upper and lower bounds were previously obtained by Khavinson-´ Swiatek, 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) = ¯ z describes an n-point gravitational lens, we determine the possible numbers of generic images.