THE ANALYSIS OF IMAGES IN N-POINT GRAVITATIONAL LENS BY METHODS OF ALGEBRAIC GEOMETRY

Albert Kotvytskiy,Vladimir Shablenko,Semen Bronza

doi:10.14311/ap.2017.57.0404

Albert Kotvytskiy, Vladimir Shablenko + Show 1 more

Open Access

https://doi.org/10.14311/ap.2017.57.0404

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Journal: Acta Polytechnica	Publication Date: Dec 30, 2017
Citations: 1	License type: CC BY 4.0

Affiliation: V. N. Karazin Kharkiv National University

Abstract

<p>This paper is devoted to the study of images in <em>N</em>-point gravitational lenses by methods of algebraic geometry. In the beginning, we carefully define images in algebraic terms. Based on the definition, we show that in this model of gravitational lenses (for a point source), the dimensions of the images can be only 0 and 1. We reduce it to the fundamental problem of classical algebraic geometry - the study of solutions of a polynomial system of equations. Further, we use well-known concepts and theorems. We adapt known or prove new assertions. Sometimes, these statements have a fairly general form and can be applied to other problems of algebraic geometry. In this paper: the criterion for irreducibility of polynomials in several variables over the field of complex numbers is effectively used. In this paper, an algebraic version of the Bezout theorem and some other statements are formulated and proved. We have applied the theorems proved by us to study the imaging of dimensions 1 and 0.</p>

Highlights

Gravitational lensing has been transformed from an effect that confirms the general theory of relativity to the research tool
The authors continue the analytic study of N-point gravitational lenses by methods of algebraic geometry [5,6,7,8]
For the model of a plane gravitational lens, we can write the equation that connects the coordinates of the source and the image coordinates, see [9, 10]

Summary

Introduction

Gravitational lensing has been transformed from an effect that confirms the general theory of relativity to the research tool. Even the cosmological parameters of the entire metagalaxy are investigated From this point of view, it seems rather strange that until now a complete analytical description has been performed only for the simplest lenses - axially symmetric lenses The authors continue the analytic study of N-point gravitational lenses by methods of algebraic geometry [5,6,7,8]. The terminology developed in algebraic geometry makes it possible to pinpoint the concept of an image in a gravitational lens. On this basis it is possible to formulate a number of statements

The physical formulation of the problem from an algebraic point of view

Conclusions