一个普适放大定理II余维5之前的通用焦散
A Universal Magnification Theorem II. Generic Caustics up to Codimension Five
摘要：
我们证明了一个定理的放大关系的所有通用的一般焦散奇点的余维5：折叠，尖点，燕尾，椭圆脐，双曲脐，蝴蝶，抛物脐，wigwam，符号脐，第二椭圆脐，第二双曲脐。具体来说，我们证明了一个通用的家庭的一般平面之间的映射表现出任何这些奇点，并为一个点的目标躺在任何地方的区域中产生的最大数量的真正的前图像（透镜图像），总有符号的放大率的前图像将始终总和为零。证明是代数的性质，并重复使用欧拉迹公式。我们还证明了一个一般的代数结果多项式，我们表明产生一个有趣的推论牛顿和反过来又容易暗示欧拉迹公式。预计将由大型综合巡天望远镜进行的宽视场成像巡天将为许多高阶焦散奇点找到观测证据。最后，由于本文的结果是一般的一般映射，而不仅仅是一般的透镜映射，预计调查结果不仅适用于引力透镜，但任何系统中，这些奇点出现。
英文摘要：
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.
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