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A universal magnification theorem. II. Generic caustics up to codimension five

J. Math. Phys. 50, 082501 (2009); doi:10.1063/1.3179163

Published 7 August 2009

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A. B. Aazami1 and A. O. Petters1,2
1Department of Mathematics, Duke University, Science Drive, Durham, North Carolina 27708, USA
2Department of Physics, Duke University, Science Drive, Durham, North Carolina 27708, USA
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, second elliptic umbilic, and second 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 preimages (lensed images), the total signed magnification of the preimages 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 also to any system in which these singularities appear. ©2009 American Institute of Physics
History: Received 15 April 2009; accepted 25 June 2009; published 7 August 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/082501/1
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KEYWORDS and PACS

Keywords
PACS
  • 98.62.Sb
    Gravitational lenses and luminous arcs
  • 95.30.Sf
    Relativity and gravitation in astrophysics
  • 98.35.-a
    Characteristics and properties of the Milky Way galaxy
  • YEAR: 2009

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ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (30)
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