An upper bound for the number of gravitationally lensed images in a multiplane point-mass ensemble

Abstract

Herein we prove an upper bound on the number of gravitationally lensed images in a generic multiplane point-mass ensemble with K planes and $g_{i}$ masses in the $i{\text {th}}$ plane. With $E_{K}$ and $O_{K}$ the sums of the even and odd degree terms respectively of the formal polynomial $\prod _{i=1}^{K}(1+g_iZ)$, the number of lensed images of a single background point-source is shown to be bounded by $E_{K}^{2}+O_{K}^{2}$. Our proof uses the theory of resultants applied to a complex variable representation of the so-called lensing map. Previous studies concerning upper bounds for point-mass ensembles have been restricted to two special cases: one point-mass per plane and all point-masses in a single plane.