How Constant Shifts Affect the Zeros of Certain Rational Harmonic Functions
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How Constant Shifts Affect the Zeros of Certain Rational Harmonic Functions Jörg Liesen1 · Jan Zur1
Received: 24 February 2017 / Revised: 13 January 2018 / Accepted: 15 January 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract We study the effect of constant shifts on the zeros of rational harmonic 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. Keywords Rational harmonic functions · Gravitational lensing · Critical curve and caustic · Cusp and fold points · Singular zeros Mathematics Subject Classification 30D05 · 31A05 · 85A04
1 Introduction The number and location of the zeros of rational harmonic functions of the form f (z) = r (z) − z,
(1.1)
Communicated by Nikos Stylianopoulos.
B
Jörg Liesen [email protected] Jan Zur [email protected]
1
TU Berlin, Institute of Mathematics, MA 3-3, Straße des 17. Juni 136, 10623 Berlin, Germany
123
J. Liesen, J. Zur
where r is a rational function, have been intensively studied in recent years. An important result of Khavinson and Neumann [5] says that if deg(r ) ≥ 2, then f may have at most 5 deg(r ) − 5 zeros. As shown by a construction of Rhie [17], this bound on the maximal number of zeros is sharp in the sense that for every n ≥ 2 there exists a rational harmonic function as in (1.1) with n = deg(r ) and exactly 5n − 5 zeros. 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 point-masses; 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 [2,6,13,14], and a comprehensive treatment can be found in the monographs [16,18]. The function modeling the gravitational point-mass lens is a special case of (1.1), namely f (z) = z − r (z), where r (z) =
n mk . z − zk
(1.2)
k=1
The poles z 1 , . . . , z n ∈ C represent the position of the respective point-masses m 1 , . . . , m n > 0 in the lens plane. For a fixed η ∈ C, a solution of f (z) = η, or equivalently a zero of f η (z) = f (z) − η, represents a lensed image of a light source at the position η in the source plane. Of great importance in this application is the behavior of the zeros under movements of the light source, i.e., changes of the parameter η. Using explicit computations, Schneider and Weiss stud
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