A Universal Magnification Theorem III. Caustics Beyond Codimension Five

TL;DR

This paper extends magnification invariants to all A, D, E caustic singularities, proving that the total signed magnification sums to zero for general plane mappings with these singularities.

Contribution

It generalizes the magnification sum invariance to an infinite family of caustic singularities beyond previous codimension limits.

Findings

Total signed magnification sums to zero for A, D, E caustic singularities.

The proof uses algebraic methods and the Euler trace formula.

Applies to mappings with maximum number of real pre-images.

Abstract

In the final paper of this series, we extend our results on magnification invariants to the infinite family of A, D, E caustic singularities. We prove that for families of general mappings between planes exhibiting any caustic singularity of the A, D, E family, and for a point in the target space lying anywhere in the region giving rise to the maximum number of lensed images (real pre-images), the total signed magnification of the lensed images will always sum to zero. The proof is algebraic in nature and relies on the Euler trace formula.