Sharp parameter bounds for certain maximal point lenses
  Starting from an $n$-point circular gravitational lens having $3n+1$ images,
Rhie (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
(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.
