DOI: https://doi.org/10.18524/1810-4215.2019.32.182518

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

O. A. Osmayev, Yu. S. Shuvalova, E. S. Bronza, K. I. Matvienko

#### Анотація

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

#### Ключові слова

gravitational lens: source image; inverse problem; complex analysis

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#### Посилання

Bronza S.D.: 2016, Scientific papers collection of UkrSURT, 160, 114.

Bronza S.D., Kotvytskiy A.T.: 2017, Bull. Kharkiv Nat. Univ. "Physics", 26, 6.

Chebotarev N. G.: 1948, Algebraic functions theory, (OGIZ, M.-L.), 396 p. (in Russian).

Dank K., Heyrovsk´ y D.: 2015, AJ, 806, 99.

Kotvytskiy A.T., Bronza S.D., Shablenko V.Yu.: 2017, Acta Polytechnica 57(6), 404.

Kotvytskiy A.T., Shablenko V.Yu., Bronza E.S.: 2018, Odessa Astron. Publ., 31, 24.

Kotvytskiy A.T., Bronza S.D., Nerushenko K. Yu., et al: 2017, VI In-

terregional Scient.-Pract. Conf. "Astronomy and Today", Vinnitsya, 198.

Osmayev O.A., Matvienko K.I.: 2018, 80-th UkrSURT SSTC theses, 216.

Witt H.J.: 1990, A&A, 236, 311.

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