The N-point gravitational lens as cover and his the profile cover N-точкова гравітаційна профіль накриття

The study of mathematical models of gravitational lenses are not direct observations. A special place in such studies is the visualization of the lens model. The image of the source and its images in the N-point gravitational lens, in the picture plane, visualizes the mathematical model - the algebraic equation of the lens. Recently, the number of studies of the equation of the N-point gravitational lens by algebraic methods has increased [6–8]. Such studies make it possible to consider the gravitational lens not only as an algebraic, but also as a topological object. In the work, the equation of the N-point gravitational lens in the complex form is studied. A bundle above the source plane is assigned to it. We investigated one subfamily of lens equations. A critical set of equations of this subfamily is a closed Jordan curve. To the equations of this subfamily we put in correspondence not only a vector bundle, but also a covering. A method for describing coverings is developed for equations whose caustic in the finite plane is a closed Jordan curve (Jordan caustic). A special case of such coverings is coverings for the equation of an N-point gravitational lens, the critical set of which is a closed Jordan curve. These equations, also, have Jordan caustics. The method is similar to the method for describing Riemann surfaces of algebraic functions, graphs ‒ profiles. The algorithm for constructing coverings and the developed method for describing these coverings illustrates an example of a cover given by a rational non-analytic function of a complex variable The covering surface has not only a Jordan caustic, but also a second-order branch point at an infinitely distant point. The methods of the theory of functions of a complex variable, algebraic geometry, algebraic topology and graph theory are used.


Introduction
In this paper, the plano N-point gravitational lens [1] is specified by the equation in a complex form and is studied as a vector bundle over the source plane.
We considered the N-point gravitational lens as a bundle above the source plane. Among other things, one subfamily of lens equations has been studied. Equations from this family are assigned not only to the vector bundle, but also to the covering map. For such equations, a method of describing covering maps has been developed.
The developed method is similar to the method of describing Riemann surfaces of algebraic functions by profiles.

Lens mapping and the equation that specifies it
The plano N-point gravitational lens determines an unequivocal complex mapping from the complex plane The mapping (1) is specified by the complex equation where n m are normalized point masses of the lens, n A are their complex coordinates [9]. Function, Denote a set of the form (2) Note that the mapping (4)  The N-point gravitational lens as cover and his the profile cover

Solution and critical set of the
Let make some remarks about the solution of the equation Equation (7) is equivalent to a system of equations: Equation (8) is equivalent to a system of polynomial equations: The system of equations (9) and therefore (8) can have one-dimensional zero-dimensionalpoint solutions, see [6]. There are algorithms that make it possible to determine a set of zero-dimensional solutions and a set of one-dimensional solutions [6]. Equation (7) and the systems of equations (8) and (9)  Let the mapping (4) be specified by equation (7), then following theorem holds.  where  is the set of solutions of the equation 0 J  (critical set). This triple is two-dimensional bundles over the complex plane; see [2].

Construction coverings
The covering for an algebraic function can be defined by a graph of a special form -a profile of a Riemann surface, for more details see [4], or a complex of segments, see [5].
We consider another case where the vector bundle can be defined as a cover, and the cover is described by a graph similar to the profile of a Riemann surface.
Let the vector bundle  be defined by equation (7).
Let, in addition, the critical set of equation (7) is a closed Jordan curve. Then the one-dimensional vector bundle over the caustic can be defined before covering. Such coverings will be called simple. If the critical set is a closed Jordan curve, then the caustic is also a closed Jordan curve. In this case, the thickness of the layer is the same at all regular points, and the onedimensional bundle can be defined before covering. Thus, in the above terms, the following theorem holds. , is an n -branched covering. Cover  is a simple cover.
A simple cover uniquely defines a gravitational lens.
The number of sheets is 4 n  .  We will call the q F  graph a wreath from q circles, and the q G contour -the main contour of the wreath. The arcs of the main contour will be called strong arcs, and loops weak arcs. This image can be considered as a directed graph. The orientation of the arcs in the П graph is induced by the orientation of the wreath arcs.

Definition. Let
In the graph of П we will distinguish two types of arcs.
Strong arcs are prototypes of arcs of the main contour. Weak arcs are prototypes of loops.
We point out some properties of the profile. Strong arcs form n of disjoint (strong) contours, q arcs in each. Weak arcs are connected only vertices from one layer. Each vertex includes two arcs: one strong and one weak. Two arcs emanate from each vertex: one strong and one weak.
The profile has an accurate coverage of n by alternating contours of the 2q length, see [4].
We accept the agreement on the profile image. We will depict a profile: -in layers (vertices from one layer are projected into your image); -by sheets (strong contours are depicted as n parallel segments); -ends of parallel segments are identified. Fig. 4 shows a profile of length 4 and thickness 3, the image of the basis loops are omitted. images to the vertices of the graphs. We assign the arcs of the graphs to the caustic arcs of their images.

The Building profiles
The arcs of the graphs gr K and gr M are defined as strong. Incidence in columns is defined as hereditary.The graph gr K has two faces. We will call the inner face of the graph gr K that face whose boundary is positively oriented. Another facet will be called external.
We define in gr M the external and internal faces.
We put in correspondence with the graph gr K r the wreath q  . We supplement gr K at the vertices with loops that lie in the outer fase of the caustic. We set the loops to positive orientation.
We supplement the graph gr M with weak arcs. The The cover over the caustic of Example 1 can be associated with the profile of the one-dimensional cover, which is shown in Fig. 3. The construction of the profile is illustrated in Fig. 2. Profile in Fig. 3. is depicted by sheets and in layers.  The two-dimensional covering, from Example 1, obviously, has a branch point of the first order at the point  . The profile of this cover is shown in Fig. 4. The cover profile, in Fig. 4 has a precise coating with alternating contours.