SOME COROLLARY FACTS OF THE N-POINT GRAVITATIONAL LENS EQUATION IN A COMPLEX FORM

Abstract

In the theory of the N-point gravita- tional lens equation, two groups of problems can be dis- tinguished. These are the so-called primal and inverse problems. Primal problems include problems of image definition in a specified lens for a specified source. In- verse problems include problems of determining a lens, source, or multiple images from one or more specified images. Inverse problem have an important applica- tions. We studied the equation of the N-point gravitational lens in a complex form. These studies became the basis for the solution of the inverse problem in the following formulation. N-point gravitational lens has specified. It is necessary to determine all other images from one of the images of a point source in N-point gravitational lens. Determine the necessary and sufficient conditions under which this problem has solutions. The algebraic formulation of the problem has the following form. The equation (of N-point gravitational lens) has specified. It is necessary to solve the problem of solutions unification (to express unequivocally all of the equation solutions through one parameter). To solve the inverse problem, we used methods of al- gebraic geometry and function theory. Branches equa- tions of any algebraic function admit unequivocal pa- rameterization by Puiseux series. The solutions of the N-point gravitational lens equation are algebraic functions defined by a certain irreducible polynomial. That polynomial has unequivocally defined by the N- point gravitational lens equation. Thus, the polyno- mial roots also admits parameterization by Puiseux se- ries. In simple cases, for lenses with a small number of point masses, the solution can be obtained in a sim- pler form. In particular, for the Schwarzschild lens and binary lens, the inverse problem has a solution in radicals.