Mathematical bases of the theory of N-point gravitational lenses. Part 1. Elements of algebraic geometry

Abstract

In this paper we consider the theory of N-point gravitational lens from the standpoint of classical algebraic geometry. The first section explains the physical statement of the problem and given the conclusion of the basic equation of the gravitational lens. In the second - a brief discussion of the main objects of study in classical algebraic geometry, and justified its application to the theory of N-point gravitational lenses. Then we give the definition of the central concepts of algebraic geometry - and the resultant theorems related. The fourth section shows, a well-known, Bezout theorem on the number of solutions of polynomial equations of the system and its corollary. In our approach, this theorem is needed to study the solutions of the gravitational lens. In the fifth section, we formulate and prove a criterion of irreducibility of polynomials in several variables over the field of complex numbers. We do not know analogues of this criterion for polynomials in several variables over a field of characteristic zero. The final section provides an overview of the solutions of systems of polynomial equations and formulated a number of challenges and problems the solution of which, in our opinion, it is advisable to apply the presented mathematical apparatus.