Sharp parameter bounds for certain maximal point lenses

Starting from an $n$ -point circular gravitational lens having $3n+1$ images, Rhie (ArXiv Astrophysics e-prints, 2003) used a perturbation argument to construct an $(n+1)$ -point lens producing $5n$ images. In this work we give a concise proof of Rhie’s result, and we extend the range of parameters in Rhie’s model for which maximal lensing occurs. We also study a slightly different construction given by Bayer and Dyer (Gen Relativ Gravit 39(9):1413–1418, 2007) arising from the $(3n+1)$ -point lens. In particular, we extend their results and give sharp parameter bounds for their lens model. By a substitution of variables and parameters we show that both models are equivalent in a certain sense.