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 ith 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_i Z), the number of lensed images of a single background point-source is shown to be bounded by E_K^2+O_K^2. 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.