Acta Polytechnica


doi:10.14311/AP.2017.57.0404
Acta Polytechnica 57(6):404–411, 2017 © Czech Technical University in Prague, 2017

available online at http://ojs.cvut.cz/ojs/index.php/ap

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

Albert T. Kotvytskiya, b, ∗, Semen D. Bronzab,
Volodymyr Yu. Shablenkoa

a Karazin Kharkov National University, Svobody Square 4, Kharkiv, 61022, Ukraine
b Ukrainian State University of Railway Transport, Feierbakh Square 7, 61050, Kharkiv, Ukraine
∗ corresponding author: kotvytskiy@gmail.com

Abstract. This paper is devoted to the study of images in N-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.
Keywords: gravitational lense, images, algebaric geometry, resultant.

1. Introduction
In modern astrophysics, gravitational lensing has been
transformed from an effect that confirms the general
theory of relativity to the research tool. Gravita-
tional lensing is used to study both stellar systems
and planets in them, and galaxies and systems of
galaxies. 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 sym-
metric lenses (see for example [1] or straight infinite
cosmic strings [2].

To analyze fairly simple 2-point gravitational lenses,
only approximate or numerical methods are used [3, 4].
In this paper, the authors continue the analytic

study of N-point gravitational lenses by methods of
algebraic geometry [5–8].

In physics, the concept of “image in a gravitational
lens” is understood intuitively and is usually not deter-
mined. However, the absence of a definition can lead
to ambiguous understanding of the concept and a dif-
ferent interpretation of some results, for example, the
theorem on the odd number of images [9, 10]. On the
other hand, 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.

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

For the model of a plane gravitational lens, we can
write the equation that connects the coordinates of the

source (the radius vector ~y) and the image coordinates
(radius vector ~x), see [9, 10]

~y = ~x−~α, (1)
where ~α is the total angle of deflection of the light
beam in the plane of the lens. In the case of an N-point
gravitational lens, the deflection angle is determined
by the following expression:

~α =
N∑

i=1
mi

~x−~li∣∣~x−~li∣∣2 , (2)
where mi are dimensionless masses whose position
in the plane of the lens is determined by the radius
vectors ~li. We have that holds

∑N
i=1 mi = 1.

Equation (1) with allowance for (2) in coordinate
form has the form


(
x1 −

N∑
i=1

mi
x1 −ai

(x1 −ai)2 + (x2 − bi)2
)
−y1 = 0,

(
x2 −

N∑
i=1

mi
x2 − bi

(x1 −ai)2 + (x2 − bi)2
)
−y2 = 0,

(3)
where ai and bi are the coordinates of the radius-vector
~li i.e. ~li = (ai,bi) .

From an algebraic point of view, system (3) is a
system of two rational equations (over a field of real
numbers) from two unknowns, which are given in
Cartesian coordinates on the R2 plane. The system
(3) will be considered, just above the field of complex
numbers C, while we denote it by (3a). The set of
solutions of system (3) is obviously the set of real solu-
tions of system (3a). We note that all the coefficients
of the equations of system (3a) are real.

404

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vol. 57 no. 6/2017 The Analysis of Images in N-point Gravitational Lens

In terms of algebraic geometry, the image of a source
in an N-point gravitational lens can be defined as
follows:

Definition. An image of a point source in an N-point
gravitational lens will be called the real solution of
system (3a) without regard to multiplicity. The set
of images is the set of different real solutions of this
system.

3. Reduction of the problem
to the fundamental problem of
classical algebraic geometry

The main problem of classical algebraic geometry
is the problem of studying systems of polynomial
equations.

Let us investigate the set of solutions of system (3a).
To do this, we transform the equations of the system
to a polynomial form

F1 = (x1 −y1)

N∏
i=1

hi −
N∑

j=1
mj (x1 −aj )

N∏
i=1,i6=j

hi = 0,

F2 = (x2 −y2)
N∏

i=1
hi −

N∑
i=1

mi(x2 −bi)
N∏

i=1,i6=j
hi = 0,

(4)

where hi = (x1 −ai)2 + (x2 − bi)2, i = 1, 2, . . . ,N.
The polynomial form of the equations in system

(4) is necessary for its investigation by methods of
algebraic geometry.
We shall consider the equations of the system (4)

over the field C of complex numbers in the affine co-
ordinate system C2. The system (4) is not equivalent
to the system (3a), but it follows from it. The set
of solutions of the system (3a) can be obtained from
the set of solutions of the system (4). For this, by
removing solutions from it in which the system (3a)
is not defined. These solutions are pairs of numbers
that are the coordinates of the point masses. Indeed,
we directly verify that the points with coordinates
(ai,bj ), i = 1, ...,N, is a solution of the system (4),
but the system (3a), in these points is not defined.
Let f1 and f2 be the left-hand sides of the first

and second equations of system (3a), M(f1,f2) be the
solution set of system (3a), V (F1,F2) be the solution
set of system (4), and Re V (F1,F2) ⊂ V (F1,F2) the
subset of its real solutions, then we have

M(f1,f2) = Re V (F1,F2)/{∪(ai,bi)}. (5)

From the theorem on the structure of the set of
solutions of a system of polynomial equations, see [11]
it follows that the set V (F1,F2) can be represented
in the form

V (F1,F2) =
(
V 0(F1,F2)

)
∪
(
V 1(F1,F2)

)
, (6)

where V 1(F1,F2) is the set of solutions depending on
a single parameter, and V 0(F1,F2) is the discrete set
of solutions of system (3a).

The set V 0(F1,F2) is obviously discrete and,
moreover, finite. The sets have dimension dim
V 0(F1,F2) = 0 and dim V 1(F1,F2) = 1.

4. Study of the set V 1(F1, F2)
(Extended solutions)

A number of theorems, which allow us to determine
if the set V 1(F1,F2) is empty, see, for example, [12,
13]. In [5], we give an algorithm that allows us to
describe this set analytically, if it is not empty. If
the set V 1(F1,F2) is not empty, then the equations of
system (3) are said to have a common component. The
equation of the common component can be obtained
from the analytical description of the set V 1(F1,F2).

4.1. 1-point lens (Schwarzschild lens)
We apply the theorems presented in the Appendix
for constructing the set V 1(F1,F2), in the case of a
single-point gravitational lens.

Let the 1-point lens have coordinates a1 = 0,b1 = 0.
Let L: R2Y → R

2
X be the transformation from the

plane of the source to the plane of the lens, determined
by the system of equations


y1 = x1 −

x1
x21 + x22

,

y2 = x2 −
x2

x21 + x22
.

(7)

Equations of the system are defined for all points
such that x21 +x22 6= 0, that is, except for the origin of
the point O(0, 0). At the origin, the inverse mapping
is not defined. But if we transform system (7) to
polynomial form{

(x21 + x
2
2)(x1 −y1) −x1 = 0,

(x21 + x
2
2)(x2 −y1) −x2 = 0,

(8)

then the inverse transformation of L−1 : R2X → R
2
Y is

completely determined by the system of equations{
x31 + x1x

2
2 −x

2
1y1 −x

2
2y1 −x1 = 0,

x21x2 + x
3
2 −x

2
1y2 −x

2
2y2 −x2 = 0.

(9)

We calculate the result R1 by the variable x1 for
which we represent the equation in lexicographic form{

x31 −y1x
2
1 + (x

2
2 − 1)x1 −x

2
2y1 = 0,

(x2 −y2)x21 + x
3
2 −x

2
2y2 −x2 = 0.

(10)

Result by degree x1 has the form

R1 =

∣∣∣∣∣∣∣∣∣∣
1 r12 r13 r14 0
0 1 r12 r13 r14
r21 0 r23 0 0
0 r21 0 r23 0
0 0 r21 0 r23

∣∣∣∣∣∣∣∣∣∣
, (11)

where r12 = −y1,r13 = x22 − 1,r14 = −x22y1,r21 =
x2 −y2,r23 = x32 −x22y2 −x2.

405



A. T. Kotvytskiy, S. D. Bronza, V. Yu. Shablenko Acta Polytechnica

We have

R1 = −x31y
2
1 + x

2
2y

2
1y2 + x2y

2
1 −x

3
2y

2
2 + x

2
2y

3
2. (12)

In order for the system equations (7) to have a
common component, we need R1 ≡ 0.
Applying Theorem 5a for the decomposition of R1

on indecomposable components, we have

x2
(
−(y21 + y

2
2 )x

2
2 + y2(y

2
1 + y

2
2 )x2 + y

2
1
)
≡ 0. (13)

The equation is divided into two equations

x2 ≡ 0,
−(y21 + y

2
2 )x

2
2 + y2(y

2
1 + y

2
2 )x2 + y

2
1 ≡ 0. (14)

Each of the equations is considered to be a polyno-
mial of the variable x2.

A polynomial is identically equal to zero if and only
if all its coefficients are equal to zero.
From here there is a system of equations{

y21 + y
2
2 = 0,

y21 = 0.
. (15)

Next we have that y1 = 0, y2 = 0. Substituting in
(7) we have

x1 −

x1
x21 + x22

= 0,

x2 −
x2

x21 + x22
= 0

⇒



x1

(
1 −

1
x21 + x22

)
= 0,

x2

(
1 −

1
x21 + x22

)
= 0.

(16)
The system (16) decomposes into three systems and

one equation: {
x1 = 0,
x2 = 0,

(17a)



x1 = 0,

1 −
1

x21 + x22
= 0,

(17b)


1 −

1
x21 + x22

= 0,

x2 = 0,
(17c)

1 −
1

x21 + x22
= 0. (18)

System (17a) has a solution x1 = 0,x2 = 0 but
this solution is not a solution of system (7), since the
system equations at the point O(0, 0) are not defined.
The system (17b) has two solutions x1 = 0, x2 =
±1.

The system (17c) has two solutions x1 = ±1, x2 =
0.
We have the transformation of equation (18) into

x21 + x
2
2 − 1 = 0. (19)

Equation (19) is the equation of an individual circle
in the plane X with a center at the point O(0, 0).

The solution of systems (17b) and (17c) satisfies
equation (19).

The solution of system (8) is the coordinates of the
points of single circle with center at the point O(0, 0).
Equation (19) is the equation of the general com-

ponent, hence the set

V 1(F1,F2) = {x1,x2 |x21 + x
2
2 − 1 = 0}. (20)

In the same way, we compute the resultant R2. By
virtue of the symmetry of variables, we have the same
solution.

4.2. 2-point lens
We research a two-point gravitational lens with equal
masses m1 = m2 = 12 .

The masses are on the abscissa at a distance a from
the origin of coordinates.

In this case, system (3) looks like this:

y1 = x1 −

1
2

x1 −a
(x1 −a)2 + x22

−
1
2

x1 + a
(x1 + a)2 + x22

,

y2 = x2 −
1
2

x2
(x1 −a)2 + x22

−
1
2

x2
(x1 + a)2 + x22

.

(21)
We transform the equation of system (21) into a

polynomial form, and represent the obtained polyno-
mials F1 and F2 in lexicographic form with increasing
degrees of variable x1:


F1 = −a2(a2 + 2x22y1)
+
(
a2 −x22 + (a

2 + x22)
)
x1 + 2y1(a2 −x22)x

2
1

+ (2x22 − 1 − 2a
2)x31 + y1x

4
1 + x

5
1,

F2 =
(
−a2x2 −x32 + (a

2 + x22)
2(x2 −y2)

)
−
(
x2 + 2(a2 −x22)(x2 −y2)

)
x21 + (x2 −y2)x

4
1,

(22)
We will remove from the system the variable x1,

using the resultant R1 = R(F1,F2).
Sylvester matrix S1 = S(F1,F2) has order (for x1)

deg F1 + deg F2 = 9.
Because R1 = det S1, we have

R1 = 4a4x22(a
2 + x22)

(
−a2y32

+ (a2y22 −y
2
2 − 4a

4y22 − 4a
2y42 )x2

+ (−4a2y2 + 4a4y2 − 4a6y2 −y21y
2

− 4a2y21y2 + 8a
4y21y2 − 4a

2y21y2

− 5y32 + 4a
2y32 − 8a

4y32 − 4a
2y52 )x

2
2

+ (−4a4 + 4a6 + y21 + 4a
2y21

− 8a4y21 + 4a
2y41 + y

2
2 − 12a

2y22

+ 8a4y22 − 8y
2
1y

2
2 − 8y

4
2 + 4a

2y42 )x
3
2

− 4(a4y2 −a2y2 −y21y2 − 2a
2y21y2

+ y41y2 −y
3
2 + 2a

2y32 + 2y
2
1y

3
2 + y

5
2 )x

4
2

+ 4(a4 − 2a2y21 + y
4
1 + 2a

2y22 + 2y
2
1y

2
2 + y

4
2 )x

5
2
)
.

(23)

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vol. 57 no. 6/2017 The Analysis of Images in N-point Gravitational Lens

In order for the system equation to have a common
component, it is sufficient that the objects R1 ≡ 0.
We have that the equation decomposes into three

simple equations and one non-trivial equation.
From the trivial equation a4 ≡ 0,x22 ≡ 0, (a2 +x22) ≡

0 it follows that their solutions are reduced to 1-lens,
or incommensurate.
We have a nontrivial equation

−a2y32 + (a
2y22 −y

2
2 − 4a

4y22 − 4a
2y42 )x2

+ (−4a2y2 + 4a4y2−4a6y2−y21y
2
2 −4a

2y21y2 + 8a
4y21y2

− 4a2y21y2 − 5y
3
2 + 4a

2y32 − 8a
4y32 − 4a

2y52 )x
2
2

+ (−4a4 + 4a6 + y21 + 4a
2y21 − 8a

4y21 + 4a
2y41 + y

2
2

− 12a2y22 + 8a
4y22 − 8y

2
1y

2
2 − 8y

4
2 + 4a

2y42 )x
3
2

− 4(a4y2 −a2y2 −y21y2 − 2a
2y21y2 + y

4
1y2

−y32 + 2a
2y32 + 2y

2
1y

3
2 + y

5
2 )x

4
2

+ 4(a4 − 2a2y21 + y
4
1 + 2a

2y22 + 2y
2
1y

2
2 + y

4
2 )x

5
2 = 0.

(24)

We equate all coefficients to zero, and have a system
of equations


−a2y32 = 0,
a2y22 −y

2
2 − 4a

4y22 − 4a
2y42 = 0,

−4a2y2 + 4a4y2 − 4a6y2 −y21y
2
2

−4a2y21y2 − 4a
2y52 − 4a

2y21y2

− 5y32 + 4a
2y32 − 8a

4y32 + 8a
4y21y2 = 0,

−4a4 + 4a6 + y21 + 4a
2y21 − 8a

4y21 + 4a
2y41 + y

2
2

− 12a2y22 + 8a
4y22 − 8y

2
1y

2
2 − 8y

4
2 + 4a

2y42 = 0,
a4y2 −a2y2 −y21y2 − 2a

2y21y2 + y
4
1y2

−y32 + 2a
2y32 + 2y

2
1y

3
2 + y

5
2 = 0,

a4 − 2a2y21 + y
4
1 + 2a

2y22 + 2y
2
1y

2
2 + y

4
2 = 0.

(25)
We have


a = 0,
y2 = 0,
−y21y

2
2 − 5y

3
2 = 0,

−y21y2 + y
4
1y2 −y

3
2 + 2y

2
1y

3
2 + y

5
2 = 0,

y41 + 2y
2
1y

2
2 + y

4
2 = 0.

(26)

The system has one solution a = 0,y1 = 0,y2 = 0.
Hence this solution reduces the 2-point gravity lens

to 1-point.
Similarly we calculate the resultant R2 as

R2 = 4a4(a−x1)x21(a + x1)
(
−a2y31 +

+ (1 + a2 + 4a4 − 4a2y21 − 4a
2y22 )y

2
1x1

+ (−4a2 + 4a4 − 4a6 + 5y21 + 4a
2y21 + 8a

4y21 − 4a
2y41

+ y22 − 4a
2y22 − 8a

4y22 − 8a
2y21y

2
2 − 4a

2y42 )y1x
2
1

+
(
4a4 + 4a6 + (−1 − 12a2 − 8a4 + 8y21 + 4a

2y21 )y
2
1

+ (−1 + 4a2 + 8a4 + 8y21 + 8a
2y21 + 4a

2y22 )y
2
2
)
x31

+ (4a2y21 + a
4y1 −y31 − 2a

2y31 + y
5
1 −y1y

2
2

+ 2a2y1y22 + 2y
3
1y

2
2 + y1y

4
2 )x

4
1

− 4(a4 − 2a2y21 + y
4
1 + 2a

2y22 + 2y
2
1y

2
2 + y

4
2 )x

5
1
)
.
(27)

We have that the solution of system (21) reduces
the 2-point gravity lens to 1-point. Whence
• for the 1-point gravitational lens set we have

V 1(F1,F2) = {x1,x2 |x21 + x
2
2 − 1 = 0};

• for the 2-point gravitational lens set we have

V 1(F1,F2) = ∅.

Based on the studies we have carried out above,
one can prove that there are no extended objects for
N-point gravitational lenses, i.e. V 1(F1,F2) = ∅.
The set M(f1,f2) can be represented in the form

M(f1,f2) = M0(f1,f2) ∪M1(f1,f2), (28)

where M0(f1,f2) = Re V 0(F1,F2)/{∪(ai,bi)} and
M1(f1,f2) = Re V 1(F1,F2)/{∪(ai,bi)}.
It is known that the set M1(f1,f2), for a point

source in 1-point lens is not empty, see for exam-
ple [9, 10, 14], coincides with V 1(F1,F2), see [5] and
is Einstein ring. But for a point source in symmetric
2-point lens, we proved [5] that the set M1(f1,f2) is
empty and put forward hypothesis: for N-point lens
this set is empty for N > 1.

5. The study of the set V 0(F1, F2)
(Point solutions)

To research the set of solutions V 0(F1,F2) of system
(3) we use the Bezout theorem, see for example [11–
13, 15].

In most monographs, the authors formulate the
Bezout theorem in geometric terms; see for exam-
ple [11, 12, 15]. One of these theorems is quoted in
Appendix.

In [13] Bezout’s theorem is formulated in algebraic
terms, but for equations given in affine coordinates.
This theorem is also quoted in Appendix.

For our purposes, we formulate this theorem in al-
gebraic terms, but for functions given in homogeneous
coordinates.

Theorem 1 (Bezout). Let G1(X0 : X1 : X2) and
G2(X0 : X1 : X2) be homogeneous polynomials,
deg G1(X0 : X1 : X2) = n, deg G2(X0 : X1 : X2) = m
and the resultant R1(G1,G2), with respect to variable
X1 not identically equal to zero. Then the resultant
R1(G1,G2) is a homogeneous polynomial with respect
to variables X0 and X2 , and deg R1(G1,G2) = n ·m.

Proof. The resultant R1(G1,G2) is a polynomial in
the variables X0 and X2. We denote it by F , and write

407



A. T. Kotvytskiy, S. D. Bronza, V. Yu. Shablenko Acta Polytechnica

F = R1(G1,G2). Let us prove that the polynomial
F = F (X0,X2) is homogeneous and of degree deg F =
n ·m. Really, we have

F(tX0, tX2) = R1
(
G1(tX0 : X1 : tX2),

G2(tX0 : X1 : tX2)
)
. (29)

According to Theorem 1a, see Appendix, we have

R1
(
G1(tX0 : X1 : tX2),G2(tX0 : X1 : tX2)

)
= det Sul

(
G1(tX0 : X1 : tX2),G2(tX0 : X1 : tX2)

)
.

(30)

where the right-hand side of expression (30) is the
determinant of the Sylvester matrix. The order of this
determinant is n + m.

Elements of the Sylvester matrix are coeficies from
the lexicographic representation of homogeneous poly-
nomials G1 and G2. We have that

G1 =
n∑

i=0
aiX

n−i
1 and G2 =

m∑
j=0

bjX
m−j
1 .

The coefficients ai = ai(X0 : X2) and bj = bj (X0 :
X2) are homogeneous polynomials.
The degree is deg ai(X0 : X2) = i and deg bj (X0 :

X2) = j, that is, ai(tX0 : tX2) = tiai(X0 : X2) and
bj (tX0 : tX2) = tjbj (X0 : X2).
We multiply every row of the determinant of the

Sylvester matrix by the parameter t in some degree.
We choose a degree so that all elements of the column
are of the same degree with respect to t.
We multiply the i-th row of the determinant of

the Sylvester matrix by ti where i = 1, 2, . . . ,m, and
j-th row by tj where j = m + 1,m + 2, . . . ,m + n.
We take the factor ts from the s-th column where
s = 1, 2, . . . ,m + n. We denote the total power of t
by S. We have

S =
m+n∑
i=1

i−
n∑

i=1
i−

m∑
i=1

i =
(n + m)(n + m + 1)

2

−
n(n + 1)

2
−
m(m + 1)

2
= nm. (31)

In this way,

det Sul
(
G1(tX0 : X1 : tX2),G2(tX0 : X1 : tX2)

)
= tnm det Sul

(
G1(X0 : X1 : X2),

G2(X0 : X1 : X2)
)
. (32)

The determinant of the Sylvester matrix does not
depend on the parameter t.
Substituting (32) in (30) and further in (29) we

have
F(tX0, tX2) = tnmF(X0,X2). (33)

Consequently, the resultant is a homogeneous polyno-
mial of degree nm.

Theorem 1 admits a generalization. We have proved
an analogous assertion for systems of equations of
several variables, see [7, 16].

We transform system (3) and apply Bezout’s theo-
rem to its study.

In the equations of system (3) we proceed to homo-
geneous coordinates. Let{

x1 = X1/X0,
x2 = X2/X0.

(34)

After reducing the equations of the system to a poly-
nomial form, we have

X2N +10 F1

(X1
X0

,
X2
X0

,y1

)
= Φ1(X0 : X1 : X2) = 0,

X2N +10 F2

(X1
X0

,
X2
X0

,y2

)
= Φ2(X0 : X1 : X2) = 0.

(35)
The coordinates X0,X1,X2 are obviously projective

coordinates.
The system (34) defines surjective mapping,

=: C2 → CP 2. The triple of complex numbers
(X0 : X1 : X2) are the coordinates of the point and
defines the point p ∈ CP 2 in the projective plane
CP 2. The triple (λX0 : λX1 : λX2) specifies the same
point if λ 6= 0.

Therefore we have the following result.

Theorem 2. The system of polynomial equations{
Φ1(X0 : X1 : X2) = 0,
Φ2(X0 : X1 : X2) = 0

(36)

has in the projective plane CP 2, counting multiplic-
ity, exactly m ·n solutions, where, m = deg Φ1, and
n = deg Φ2, if gcd(Φ1, Φ2) belongs to the coefficient
field C.

Functions Φ1 = Φ1(X0 : X1 : X2) and Φ2 = Φ2(X0 :
X1 : X2) are homogeneous functions of degree 2N + 1.
If, at least one of the coordinates of the point p is

equal to zero, say that this point is irregular. Other-
wise, the point is called regular.

A straight line that consists of irregular points is
called an irregular line.
The projective plane CP 2 has three irregular

straight lines, which are given by the equations

X0 = 0, X1 = 0, X2 = 0. (37)

The set of points CP 2 one of the coordinates, which
is equal to the number h 6= 0, is called affine map
on CP 2 and denoted by A2(h). The complement of
a CP 2\A2(h) consists of a one-dimensional complex
projective subspace, which is called an infinitely dis-
tant line of the affine map , see for example [11, 15].
The infinitely distant line of any affine map A2(h) is
evidently irregular.
In particular, if we put X0 = 1, then the set of

points CP 2 with coordinates (1 : X1 : X2) will be

408



vol. 57 no. 6/2017 The Analysis of Images in N-point Gravitational Lens

affine map of A2(1), and the infinity of the straight
line of this map will be given by equation X0 = 0.
Consider the situation of general position, i.e. the

source is not on the caustic. In this case, the Jacobian
of the system of lens equations is not equal to zero.
Theorem 3. In a situation of general position (the
Jacobian of the system of lens equations is not equal
to zero), the number of point images in an N-point
gravitational lens has parity opposite to the parity of
the number N.

In the proof of Theorem 3 we use the following
lemma.
Lemma 1. The number of irregular solutions of sys-
tem (37) , on line X0 = 0, is 2N.
Proof. Using (37), we reduce the system to the form


(X1 −X0y1)
N∏

i=1
Hi

−X20
N∑

j=1
mj (X1 −X0aj )

N∏
i=1,i6=j

Hi = 0,

(X2 −X0y2)
N∏

i=1
Hi

−X20
N∑

j=1
mj (X2 −X0bj )

N∏
i=1,i6=j

Hi = 0,

(38)

where Hi = (X1 −X0ai)2 + (X2 −X0bi)2.

Let X0 = 0. We have

X1

N∏
i=1

(X21 + X
2
2 ) = 0,

X2

N∏
i=1

(X21 + X
2
2 ) = 0

⇒

{
X1(X21 + X

2
2 )

N = 0,
X2(X21 + X

2
2 )

N = 0
⇒ (X21 + X

2
2 )

N = 0

⇒ X1 = ±iX2 ⇒
{
X1 = c,
X2 = ±ic.

(39)

Finally we have two N-fold solutions: P1 = (0 : a :
ic) and P2 = (0 : a : −ic).

Proof of Theorem 2. For the degrees of the polynomi-
als of systems (3) and (5) we have deg F1 = deg F2 =
deg Φ1 = deg Φ2 = 2N + 1.

By Bezout’s theorem, the system of equations (36)
has (2N + 1)2 solutions, which include an even num-
ber of 2q complex conjugate solutions and P = 2N
irregular solutions.
Therefore, the number of real solutions of system

(36),

card
(
real V 0(F1,F2)

)
= (2N + 1)2 − 2q −P

= (2N + 1)2 − 2q − 2N
= 4N2 + 2N + 1 − 2q. (40)

From the fact that the restriction of the inverse
mapping =−1 : CP 2 → C2 to the affine map A2(1)
is a bijection that is given by the equations X0 = 1,
X1 = x1, X2 = x2, we have

card
(
M0(f1,f2)

)
= card

(
real V 0(F1,F2)

)
−N

= 4N2 + N + 1 − 2q. (41)

In a situation of general position, the point source
is not on the caustic, therefore, all elements of the set
real V (f1,f2) are different.
In this case, each point of the set real V (f1,f2) is,

by definition, an image.
It follows from (9) that the parity of the number of

images is opposite to the parity of the number N.

Theorem 3 does not contradict the theorem on
the oddness of the number of images in transparent
lenses [9, 10].

Example 1. For a 1-point lens, the number of images
is 2, see [9, 10, 14].

Example 2. For a 2-point lens, the number of images
is 3 or 5 see [17].

6. Conclusions
Applying methods of algebraic geometry, we con-
structed an algorithm that separates images of di-
mensions 1 and 0.
In the present paper, for an image of dimension 1,

it is proved that for single-point sources there exists
a unique image of dimension 1-the Einstein ring; Ein-
stein’s ring is only in a single-point lens; the point
source in other lenses does not have images of dimen-
sion 1 for N > 1. For an image of dimension 0, it is
proved that in any N-point gravitational lens: there
are a finite number of images; the parity of the number
of images is always the opposite of the parity of the
number N.

The assertion for the number of images of dimension
0 was proved by us earlier, see [18]. In [18], we used the
geometric method of algebraic geometry-the Newton
diagram. In the present paper all the assertions are
proved algebraically. This opens the possibility, to use
N-point gravitational lenses, not only approximate
or numerical methods, but also computer algebra
systems.

A. Appendix
Let f(x,y) be a function of two variables, and f(x,y),
at the point (x0,y0), n-times continuous, differentiable
function. Then Taylor’s formula holds:

f(x,y) = f(x0,y0) +
n∑

k=1
f(k)(x−x0,y −y0)

+ rn(x,y), (42)

409



A. T. Kotvytskiy, S. D. Bronza, V. Yu. Shablenko Acta Polytechnica

where

f(k)(x−x0,y −y0)

=
n∑

k=1
Cik

∂kf(x0,y0)
∂xk−i∂yi

(x−x0)k−i(y −y0)i, (43)

and f(x,y) is the remainder term.
If f(x,y) is a polynomial, and m = deg f(x,y), then

rn(x,y) = 0 for all n ≥ m.

Definition 1a. We say that a pair of numbers x0,y0
is a s-multiple solution of equation f(x,y) = 0 if:
• f(x0,y0) = 0;
• f(i)(x−x0,y −y0) ≡ 0, i = 1, 2, . . . ,s− 1;
• f(s)(x − x0,y − y0) 6= 0, s ≤ n in some neighbor-

hood of the point (x0,y0); we will write this fact as
mult(f(x0,y0)) = s.

For example point (0, 0) is s-multiple solution of
equation f(x,y) = 0 if:
• f(0, 0) = 0;
• f(i)(x,y) ≡ 0, i = 1, 2, ...,s− 1;
• f(s)(x,y) 6= 0, s ≤ n in some neighborhood of the
point (0, 0).

Such a solution is called a s-multiple zero solution.

Definition 2a. Let the pair of numbers x0,y0 be a
solution of the system of equations{

f(x,y) = 0,
g(x,y) = 0

(44)

and q = min(mult(f(x0,y0)), mult(g(x0,y0))). The
solution x0,y0 will be called q-multiple of the solution
of the system of equations (44), while we will write
q = min(mult(f,g)(x0,y0)).

The concept of a multiple solution of a system
of equations can obviously be extended to systems
with an arbitrary number of equations from several
variables.

The resultant of polynomials is one of the basic
concepts of classical algebraic geometry.
In the modern literature [12, 19, 20], the resultant

of polynomials is usually defined as follows.

Definition 3a. Let K-arbitrary field, f(x) and g(x)
polynomials in K[x]. The resultant R(f,g) of polyno-
mials f(x) and g(x) is called an element in the field
K, defined by the formula

R(f,g) = an0 b
m
0

n∏
i=0

m∏
j=0

(αi −βj ), (45)

where αi,βi are roots of polynomials f(x) =∑n
i=0 aix

n−i and g(x) =
∑m

j=0 bjx
m−j, correspond-

ingly, with the highest coefficients, a0, b0 such that
a0 6= 0, b0 6= 0.

Let the roots of the polynomials f(x) and g(x)
be known. To calculate their resultant, one can use
formula (45).

If we know only the coefficients of these polynomials,
then we can use the Sylvester matrix to calculate their
resultant. The Sylvester matrix is a block matrix of
two blocks. Each block has one ribbon matrix. We
have a definition of the Sylvester matrix.

Definition 4a. Matrix Sylvester for polynomials
f(x) =

∑n
i=0 aix

n−i and g(x) =
∑m

j=0 bjx
m−j, we

call a square matrix S = S(f,g) of order n + m with
elements sij defined by the formula

sij =




aj−i, if 0 ≤ j − i ≤ n,
i = 1, . . . ,m, j = 1, . . . ,n + m,

bj−i+m, if 0 ≤ j − i + m ≤ n,
i = m + 1, . . . ,m + n,
j = 1, ...,n + m,

0, for other i,j,

(46)

i.e.,

S(f,g) = [sij ] =

n rows




m rows







a0 a1 · · · an 0 · · · 0
0 a0 a1 · · · an 0 0

... ... ...
0 · · · 0 a0 a1 · · · an
b0 b1 · · · bm 0 · · · 0
0 b0 b1 · · · bm 0 0

... ... ...
0 · · · 0 b0 b1 · · · bm



. (47)

The resultant polynomials R(f,g) and the Sylvester
matrix Sul(f,g) are connected by the following theo-
rem.

Theorem 1a. The resultant R(f,g) of the polyno-
mials f and g is equal to the determinant of Sylvester
matrix these polynomials, i.e., R(f,g) = S(f,g).

For the proof see, e.g., [19, 20].

Theorem 2a. Polynomials f and g have a common
root if and only if

R(f,g) = 0. (48)

For the proof see, e.g., [13].

Theorem 3a (Bezout). The number of intersection
points of plane curves Φ1 and Φ2 (counted taking
into account the multiplicity) is equal to nm, where
m = deg Φ1, and n = deg Φ2, if the curves:
• do not have common components;
• are defined over an algebraically closed field;
• are considered on the projective plane.

For the proof see, e.g., [12].

410



vol. 57 no. 6/2017 The Analysis of Images in N-point Gravitational Lens

Theorem 4a. Let f(x1,x2) =
∑i+j≤n

i,j=0 fijx
i
1x

j
2 and

g(x1,x2) =
∑i+j≤m

i,j=0 gijx
i
1x

j
2 be polynomials. Let

their coefficients be such that fn0 6= 0, f0n 6= 0,
gm0 6= 0, g0m 6= 0. Then

deg R1
(
f(x1,x2),g(x1,x2)

)
= deg f(x1,x2) deg g(x1,x2) = nm. (49)

For the proof see, e.g., [12].

Theorem 5a. The polynoms F1(x1,x2), F2(x1,x2)
have a non-trivial common component if and
only if R1(F1(x1,x2),F1(x1,x2)) or R2(F1(x1,x2),
F1(x1,x2)).

For the proof see, e.g., [12].

Definition 5a. A formal sum G = G(x1,x2) of the
form G =

∑i+j≤n
i,j=0 gijx

i
1x

j
2 is called a polynomial n-

form of variables x1,x2 over the field K. that is, G
is a polynomial of degree n in variables x1,x2 with
indefinite coefficients gij in the field K.

The expression “the function will be sought in the
form of a n-form” is usually understood as a procedure
for determining the undetermined coefficients of a
given n-form.

Theorem 6a (Criterion of non-decomposability). Let
F be a polynomial in the variables over the field of
complex numbers and . Let , and let be the -form
of the variables x1 and x2 over the field of complex
numbers. Let and be the resultants with respect to
the variables and respectively.

The polynomial F is not decomposable if and only
if the systems of equations

∂i

∂xi1
R1
(
F(x1,x2),G(x1,x2)

)
= 0,

∂i

∂xi2
R2
(
F(x1,x2),G(x1,x2)

)
= 0, (50)

where i = 1, ...,m, m = deg R1(F,G), have only zero
solutions.

The system (50) is considered with respect to the
undetermined coefficients gij n-form G, as with respect
to unknown variables.

For the proof see, e.g., [7].
The theorem admits a generalization to the case of

systems of equations of several variables see [7, 16].
The criterion was formulated and proved by the

authors earlier [7].

References
[1] E. Y. Bannikova, A. T. Kotvytskiy. Three Einstein
rings: explicit solution and numerical simulation.
MNRAS 445(4):4435–4442, 2014.
doi:10.1093/mnras/stu2068.

[2] A. T. Kotvytskiy. Gravitational lensing by straight
cosmic strings. TMP 184(1):160–174, 2015.
doi:10.1007/s11232-015-0315-x.

[3] A. Cassan. An alternative parameterisation for
binary-lens caustic-crossing events. Astronomy and
Astrophysics 491(2):587–595, 2008.
doi:10.1051/0004-6361:200809795.

[4] P. Schneider, A. Weiss. The two-point-mass lens:
detailed investigation of a special asymmetric
gravitational lens. Astronomy and Astrophysics
164(2):237–259, 1986.

[5] A. T. Kotvytskiy, S. D. Bronza, K. D. Nerushenko,
V. Y. Shablenko. Matematychnii zmist kiltsia
Einshteina ta umovy yogo vynyknennja. doslidzhennya
uzahal’nenyh umov[in ukrainian]. In Zbirnyk naukovyx
prac’ VI-i Mizhrehional’noi naukovo-praktychnoi
konferencii "Astronomiya i s’ogodennya", pp. 198–213.
Vinnytsia, 2017.

[6] A. T. Kotvytskiy, S. D. Bronza, S. R. Vovk.
Estimating the number of images N-point gravitational
lenses algebraic geometry methods. Visnyk KhNU,
serija "Fizyka" 24:55–59, 2016.

[7] S. D. Bronza, A. T. Kotvytskiy. Mathematical bases
of the theory of N-point gravitational lenses. Part 1.
Elements of algebraic geometry. Visnyk KhNU, seriya
"Fizyka" 26:6–32, 2017.

[8] A. T. Kotvytskiy, S. D. Bronza. Mathematical bases
of the theory of N-point lenses. Odessa Astronomical
Publications 29:31–33, 2016.
doi:0.18524/1810-4215.2016.29.84958.

[9] A. F. Zakharov. Gravitacionnye linzy i mikrolinzy [in
Russian]. Janus-K, Moscow, 1997.

[10] P. Schneider, J. Ehlers, E. Falco. Gravitational lenses.
Second printing. Springer-Verlag Berlin Heidelberg, 1999.

[11] I. V. Arzhantcev. Bazisy Grebnera i sistemy
algebraicheskih uravnenij [in Russian]. MCNMO,
Moscow, 2003.

[12] R. J. Walker. Algebraic curves. Springer-Verlag New
York, 1978.

[13] E. Kalinina, A. J. Uteshev. Teoria iskluchenij [in
Russian]. NII chimii SPbGU Saint Petersburg, 2002.

[14] P. V. Bliokh, A. A. Minakov. Gravitational Lenses
[in Russian]. Naukova Dumka, Kiev, 1989.

[15] M. Reid. Undergraduate Algebraic Geometry. Math
Inst., University of Warwick, 2013.

[16] S. D. Bronza. Kriteriy neprivodimosti mnogochlenov
ot dvuh peremennyih nad polem kompleksnih chisel[in
russian]. In Zbirnyk naukovyx prac’, pp. 114–115.
UkrDUZT, 2016.

[17] S. H. Phie. Infimum microlensing amplification of the
maximum number of images of N-point lens systems.
The astrophysical journal 484(1):63–69, 1997.
doi:10.1086/304336.

[18] A. T. Kotvytskiy, S. D. Bronza, V. Y. Shablenko.
Correlation of the number of images of an N-point
gravitational lens and the number of solutions of its
system. In Astronomy and beyond: astrophysics,
cosmology, cosmomicrophysics,astroparticle physics,
radioastronomy and astrobiology, p. 12. Odessa
International Astronomical Gamow Conference-School,
2017.

[19] B. L. van der Waerden. Modern Algebra I, II. New
York:Frederic Ungar Publishing Co., 1950.

[20] S. Lang. Algebra. Revised Third Edition. Columbia
University. New York, 1965.

411

http://dx.doi.org/10.1093/mnras/stu2068
http://dx.doi.org/10.1007/s11232-015-0315-x
http://dx.doi.org/10.1051/0004-6361:200809795
http://dx.doi.org/0.18524/1810-4215.2016.29.84958
http://dx.doi.org/10.1086/304336

	Acta Polytechnica 57(6):404–411, 2017
	1 Introduction
	2 The physical formulation of the problem from an algebraic point of view
	3 Reduction of the problem to the fundamental problem of classical algebraic geometry
	4 Study of the set V1(F1,F2) (Extended solutions)
	4.1 1-point lens (Schwarzschild lens)
	4.2 2-point lens

	5 The study of the set V0(F1,F2) (Point solutions)
	6 Conclusions
	A Appendix
	References