Euclidean Algorithm for a Gravitational Lens in a Polynomial Equation

Abstract

The Euclidean algorithm in algebra is applied to a class of gravitational lenses for which the lens equation consists of any set of coupled polynomial equations in the image position. In general, this algorithm allows us to reduce an apparently coupled system to a single polynomial in one variable (say $x$ in Cartesian coordinates) without the other component (say $y$), which is expressed as a function of the first component. This reduction enables us to investigate the lensing properties in an algebraic manner: For instance, we can obtain an analytic expression of the caustics by computing the discriminant of the polynomial equation. To illustrate this Euclidean algorithm, we re-examine a binary gravitational lens and show that the lens equation is reduced to a single real fifth-order equation, in agreement with previous works. We apply this algorithm also to the linearlized Kerr lens and find that the lens equation is reduced to a single real fifth-order one.

8 pages (PTPTeX); accepted for publication in Prog. Theor. Phys

References in corpus (6)

• Can we see a rotating gravitational lens?
• The magnification invariant of simple galaxy lens models
• Separability of Rotational Effects on a Gravitational Lens
• Images for a Binary Gravitational Lens from a Single Real Algebraic Equation
• Images for an Isothermal Ellipsoidal Gravitational Lens from a Single Real Algebraic Equation
• Algebraic Properties of the Real Quintic Equation for a Binary Gravitational Lens

Cited by in corpus (4)

• The Chang-Refsdal Lens Revisited
• Beyond Caustic Crossings: Properties of Binary Microlensing Light Curves
• Perturbation theory of N point-mass gravitational lens systems without symmetry: small mass-ratio approximation
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