Counting Zeros of Harmonic Rational Functions and Its Application to Gravitational Lensing

TL;DR

This paper investigates the maximum and minimum number of solutions to harmonic rational equations relevant to gravitational lensing, showing that all intermediate counts are achievable under generic conditions.

Contribution

It characterizes the possible numbers of solutions to harmonic rational equations in gravitational lensing, demonstrating the realizability of all intermediate counts within bounds.

Findings

All numbers of solutions within bounds are generically attainable.

The study provides bounds for solutions to harmonic rational equations.

Application to gravitational lensing determines possible image counts.

Abstract

General Relativity gives that finitely many point masses between an observer and a light source create many images of the light source. Positions of these images are solutions of $r(z)=\bar{z},$ where $r(z)$ is a rational function. We study the number of solutions to $p(z) = \bar{z}$ and $r(z) = \bar{z},$ where $p(z)$ and $r(z)$ are polynomials and rational functions, respectively. Upper and lower bounds were previously obtained by Khavinson-\'{S}wi\c{a}tek, Khavinson-Neumann, and Petters. Between these bounds, we show that any number of simple zeros allowed by the Argument Principle occurs and nothing else occurs, off of a proper real algebraic set. If $r(z) = \bar{z}$ describes an $n$-point gravitational lens, we determine the possible numbers of generic images.