A Universal Magnification Theorem II. Generic Caustics up to Codimension Five

TL;DR

This paper proves a universal magnification relation for all generic caustic singularities up to codimension five, showing that the total signed magnification of pre-images always sums to zero, with broad implications for lensing and other systems.

Contribution

It establishes a comprehensive algebraic proof of magnification relations for complex caustic singularities up to codimension five, extending previous results to more intricate singularities.

Findings

Total signed magnification sums to zero for all considered singularities.

Provides algebraic proof using Euler trace formula and polynomial properties.

Applicable to gravitational lensing and other systems with similar singularities.

Abstract

We prove a theorem about magnification relations for all generic general caustic singularities up to codimension five: folds, cusps, swallowtail, elliptic umbilic, hyperbolic umbilic, butterfly, parabolic umbilic, wigwam, symbolic umbilic, 2nd elliptic umbilic, and 2nd hyperbolic umbilic. Specifically, we prove that for a generic family of general mappings between planes exhibiting any of these singularities, and for a point in the target lying anywhere in the region giving rise to the maximum number of real pre-images (lensed images), the total signed magnification of the pre-images will always sum to zero. The proof is algebraic in nature and makes repeated use of the Euler trace formula. We also prove a general algebraic result about polynomials, which we show yields an interesting corollary about Newton sums that in turn readily implies the Euler trace formula. The wide field imaging surveys slated to be conducted by the Large Synoptic Survey Telescope are expected to find observational evidence for many of these higher-order caustic singularities. Finally, since the results of the paper are for generic general mappings, not just generic lensing maps, the findings are expected to be applicable not only to gravitational lensing, but to any system in which these singularities appear.