How Constant Shifts Affect the Zeros of Certain Rational Harmonic Functions

Abstract

We study the effect of constant shifts on the zeros of rational harmomic functions f (z) = r(z) − z. In particular, we characterize how shifting through the caustics of f changes the number of zeros and their respective orientations. This also yields insight into the nature of the singular zeros of f . Our results have applications in gravitational lensing theory, where certain such functions f represent gravitational point-mass lenses, and a constant shift can be interpreted as the position of the light source of the lens.

The maximum number of zeros of $r(z) - \overline{z}$ revisited

“…Let f be the Rhie function from [11] (see also [12,13]), which has a rational function r of the type (n − 1, n), and which has 5n − 5 zeros. The results in [8] imply that for sufficiently small |c| > 0 the function f c has the same number of zeros as f . The corresponding rational function r c then is of the type (n, n), so the bound N (f ) ≤ 5n − 5 is sharp also in the case (n, n).…”

“…A second reason why we have formulated the result for the function f c (rather than just f ) is the application of rational harmonic functions in the context of gravitational lensing; see [5] for a survey. In that application a constant shift represents the position of the light source of the lens, and the change of the number of zeros under movements of the light source is of great interest; see, e.g., [8] for more details. Our note is organized as follows.…”

Number and location of pre-images under harmonic mappings in the plane

“…For these functions, the number of zeros or pre-images is intensively studied; see e.g. [38,19,17,13,7,25,26,32,20,21,22,5,16].…”

“…Here we focus on solutions of f (z) = η for given (but arbitrary) η ∈ C. As shown in [21] for rational harmonic mappings of the form f (z) = r(z) − z, the number of solutions can vary significantly under changes of η. Moreover, changes only occur when η is "moved" through the caustics of f ; see Figure 1.…”

A Newton method for harmonic mappings in the plane

“…We refer to the expository articles [22,34], and the survey [3]. More recent articles on harmonic mappings with applications to gravitational lensing are [4,20,29,37,38,27,25].…”