How Cumbersome is a Tenth Order Polynomial?: The Case of Gravitational Triple Lens Equation

Abstract

Three point mass gravitational lens equation is a two-dimensional vector equation that can be embedded in a tenth order analytic polynomial equation of one complex variable, and we can solve the one variable equation on the source trajectories using recipies for Fortran or $C$ (portable for $C$++ or $C_{jj}$) in Numerical Recipes, or using packages such as Mathemetica, Matlab, etc. This ready solvability renders fitting microlensing light curves including triple lenses a normal process, and such was done in a circumbinary planet fit for MACHO-97-BLG-41. Subsequently, there was a claim that converting the triple lens equation into the analytic equation was rather cumbersome, and the impressionable judgement has caused an effect of mysterious impedance around the perfectly tractable lens equation. There are judgements. Then, there is nature. We looked up for one of the quantities of highest precision measurements: electron $g$-factor correction $a_e \equiv g/2-1$. The current best experimental values of $a_e$ agree to eight significant digits with the theoretical value, and the theoretical calculation involves more than one thousand Feynman diagrams -- many orders of magnitude messier than the triple lens equation coefficients. We seem to have only choice to be compliant to nature and its appetite for elegant mess and precision numerics. In fact, the triple lens equation coefficients take up less than a page to write out and are presented here for users' convenience.