Euclidean Algorithm for a Gravitational Lens in a Polynomial Equation

The Euclidean algorithm in algebra is applied to a class of gravitationallenses for which the lens equation consists of any set of coupled polynomialequations in the image position. In general, this algorithm allows us to reducean apparently coupled system to a single polynomial in one variable (say $x$ inCartesian coordinates) without the other component (say $y$), which isexpressed as a function of the first component. This reduction enables us toinvestigate the lensing properties in an algebraic manner: For instance, we canobtain an analytic expression of the caustics by computing the discriminant ofthe polynomial equation. To illustrate this Euclidean algorithm, we re-examinea binary gravitational lens and show that the lens equation is reduced to asingle real fifth-order equation, in agreement with previous works. We applythis algorithm also to the linearlized Kerr lens and find that the lensequation is reduced to a single real fifth-order one.Comment: 8 pages (PTPTeX); accepted for publication in Prog. Theor. Phy