Sharp parameter bounds for certain maximal point lenses

Abstract

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.

Perturbing Rational Harmonic Functions by Poles

“…Moreover, if r > 0 and ε > 0 are sufficiently small, then there exist further 3d zeros corresponding to the 3d zeros of R 0 (z) − z away from z 0 = 0. The results in [16] show that the same perturbation behavior can be observed for Rhie's original function (1 − ε)R 0 (z) + ε z − z. A numerical example for this function with d = 7 is shown in Fig.…”

“…An elementary proof and sharp bounds on the parameters r and ε that guarantee extremality of R ε (z) are given in [16]. Fig.…”

“…While the function R ε (z) considered by Rhie arises from a convex combination of R 0 (z) with the pole 1 z , it was shown in [16] that qualitatively the same "zero creating effect" can be observed for a purely additive perturbation with 1 z , resulting in the function R 0 (z) + ε z − z. The analyses of this effect draw heavily from the rotational symmetry of R 0 (z).…”

How Constant Shifts Affect the Zeros of Certain Rational Harmonic Functions

“…for some ρ > 0. Functions of this form have been frequently studied in the context of gravitational lensing; see, e.g., the original work of Mao, Petters and Witt [12], and the more recent articles [10,20], which contain many further references. We choose n = 3 and ρ = 3 5 , and plot the zeros of f η for several constant shifts η in Figure 8.…”

“…Several authors have derived more refined bounds on the maximal number of zeros which depend on the degrees of the numerator and denominator polynomials of r; see, e.g., [8] and the references given there. Rhie made her construction in the context of astrophysics, where certain rational harmonic functions model gravitational lenses based on n pointmasses; see the Introduction of [21] for a brief summary of Rhie's construction, and [10] for a detailed analysis. Descriptions of the connection between complex analysis and gravitational lensing are given, for example, in the articles [2,6,13,15], and a comprehensive treatment can be found in the monographs [14,18].…”