Elimination method for construction images in N-point gravitational lenses for circular source

Gravitational lenses have become valuable tools for studying astrophysical objects, but analytical methods for analyzing gravitational lenses are still underdeveloped, particularly in the construction of images. Although analytical methods exist for single-point and axially symmetric lenses, there is a lack of exact analytical solutions for general gravitational lenses, leading to the frequent utilization of approximate methods. In this work, we aim to advance precise analytical methods for constructing images from circular sources in gravitational lenses. We begin by formulating the gravitational lens equations using the general theory of relativity. By introducing dimensionless variables, we express the lens equation in a convenient form. We then explore the mathematical concepts of resultant and Sylvester matrix, which provide tools for eliminating variables and finding common roots of polynomials. We apply these concepts to eliminate the parameter in the lens equations and obtain expressions for the image coordinates. We demonstrate the application of our method for a one-point lens and a binary lens. For a one-point lens, we derive an equation that determines the images of a circular source. We analyze special cases where the source is at the origin, touches the origin, or is arbitrarily positioned. In the general case of a binary lens, an expression is derived that describes all images resulting from a circular source located at any arbitrary position. Our method offers a precise analytical approach to construct images from circular sources in gravitational lenses. By advancing analytical techniques, we reduce the reliance on numerical simulations, which can be computationally intensive and may not guarantee the discovery of all solutions. This work contributes to the understanding and characterization of gravitational lenses and provides a foundation for further developments in the field.