  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.
