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

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 ∏ _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.