A Survey on the Maximal Number of Solutions of Equations Related to Gravitational Lensing

Abstract

This paper is a survey of what is known about the maximal number of solutions of the equation f(z)=zˉ,f(z) = \bar {z}, in particular when f is the Cauchy transform of a compactly supported positive measure. When f is a rational function, the number of solutions of this equation is equal to the number of images seen by an observer of a single light source deflected by a gravitational lens (such as a galaxy). We will discuss what is known in the context of harmonic polynomials, rational functions, polynomials in z and zˉ\bar {z} (but not harmonic!) and even transcendental functions that arise in situations involving continuous mass distributions for different shapes. In particular, we discuss an example related to the lens equation for a limaçon-shaped gravitational lens.

Full text

A Survey on the Maximal Number of Solutions
of Equations Related to Gravitational Lensing
Catherine Bénéteau and Nicole Hudson
Abstract This paper is a survey of what is known about the maximal number of
solutions of the equation f.z/DNz;in particular when fis the Cauchy transform of
a compactly supported positive measure. When fis a rational function, the number
of solutions of this equation is equal to the number of images seen by an observer
of a single light source deflected by a gravitational lens (such as a galaxy). We will
discuss what is known in the context of harmonic polynomials, rational functions,
polynomials in zand Nz(but not harmonic!) and even transcendental functions that
arise in situations involving continuous mass distributions for different shapes. In
particular, we discuss an example related to the lens equation for a limaçon-shaped
gravitational lens.
Keywords Cauchy integrals • Gravitational lensing • Harmonic polynomials •
Schwarz function
2010 Mathematics Subject Classification 30E10, 30E20, 85-02
1 Introduction
Studying and counting roots of systems of equations has been an important question
in mathematics for centuries. In the complex plane, the Fundamental Theorem of
Algebra, whose first proofs (with some gaps) are usually attributed to d’Alembert
in 1746 and Gauss in 1799, tells us that every polynomial of degree nof a single
complex variable zhas exactly ncomplex roots (counting multiplicity). However,
much less is known about the number of roots of polynomials containing both zand
Nz. A special case of interest is that of harmonic polynomials, that is, functions of
the type p.z/Cq.z/,wherepand qare complex polynomials. As an example of the
many open problems in this area, it is still not known what the maximum number of
roots of the equation p.z/DNz2is, if pis an arbitrary complex polynomial of degree
C. Bénéteau () • N. Hudson
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
e-mail: cbenetea@usf.edu;nicolehudson@mail.usf.edu
© Springer International Publishing AG 2018
M. Agranovsky et al. (eds.), Complex Analysis and Dynamical Systems,
Trends in Mathematics, https://doi.org/10.1007/978-3-319-70154-7_2
23

24 C. Bénéteau and N. Hudson
Fig. 1 Four separate images
of a single quasar located
behind a galaxy, with an
additional image too dim to
see. Credit: NASA, ESA, and
STScI. Source: http://
hubblesite.org/image/22/
news_release/1990-20
n>3. If we extend this realm of questioning to include rational functions, some of
these mathematical problems become questions in gravitational microlensing.
Indeed, astrophysicists have long known the equation that relates the position of
a light source and the mass distribution of a gravitational lens (e.g., a galaxy) to
the images of the source formed by the lens (see Fig. 1). This equation depends on
the mass distribution of the lens and can be written in complex form in such a way
that positions and numbers of roots of rational functions occurring in the equation
correspond precisely to those of the images formed by the gravitational lens.
Over the last 15 years, there have been many articles published investigating
these questions from a mathematical point of view. In spite of this, there are still
many remaining open questions. This paper is a survey of some of the known results
and conjectures related to counting numbers of solutions of equations of this type.
We will be particularly interested in harmonic polynomials, rational functions aris-
ing from gravitational lenses consisting of point masses, non-harmonic polynomials
in zand Nz, and certain transcendental functions that arise in situations involving
continuous mass distributions for different shapes. In particular, we discuss an
example related to the lens equation for a limaçon-shaped gravitational lens.
In Sect. 2, we introduce polynomials in zand Nzand discuss the particular case
of harmonic polynomials together with known results and open problems. We then
turn to the case of rational functions and explore the connection with gravitational
lensing. In Sect. 3, we address the case of continuous distributions: we first
examine what is known for gravitational lenses that are ellipses with uniform mass
distributions; we then examine a similar situation for a limaçon-shaped gravitational
lens. Our main contribution there is the lens equation in that case, together with
upper estimates on number of images produced. Although the estimates are not at
all sharp in that context, it may be that this approach will provide a model that is
useful for numerics and may offer insight into appropriate conjectures for future

A Survey on the Maximal Number of Solutions of Equations Related to... 25
research. In closing, we survey some results related to isothermal densities, which
give rise to lens equations involving transcendental functions.
Thanks and Dedication The authors would like to thank Dmitry Khavinson for
many helpful discussions during the writing of this paper. The first author would
also like to dedicate this paper to her dear friend Sasha Vasiliev, a wonderful
mathematician and person, who is dreadfully missed.
2 Harmonic Polynomials, Rational Functions, and Lensing
2.1 Polynomials in z and Nz
The Fundamental Theorem of Algebra counts the exact number of (complex) roots
of a complex polynomial of one variable in terms of the degree of the polynomial;
however, if the polynomial involves the complex variable zand its conjugate Nz,
the situation immediately becomes much more complicated. Consider, for complex
numbers ak;j;the polynomial equation
f.z;Nz/D
n
X
kD0
m
X
jD0
ak;jzkNzjD0:
One (perhaps naive) approach is to rewrite fin terms of the real variables xand y,
where zDxCP{y:This results in
f.z;Nz/Df.xCP{y;xP{y/
D
nCm
X
iD0
nCm
X
lD0
ci;lxiylDg.x;y/:
Separating the equation g.x;y/D0into its real and imaginary parts yields a system
of two real equations in xand y, and counting the common intersection points (in
the real plane) of this system becomes a problem in algebraic geometry.
Notice that it is not always the case that there are any solutions at all or that
number of roots is finite! Take the simple example zNzD1, which of course gives
the whole unit circle as a solution, while the equation zNzD1has no solutions.
On the other hand, consider the following trivial example: the polynomial f.z;Nz/D
z4P{NzC5C3P{can be rewritten as g.x;y/Dx4yC5CP{. y4xC3/; which
gives rise to a system of two linear equations:
0Dx4yC50Dy4xC3:
This system can be immediately solved and leads to a single complex root zD
1
15 .17 C23P{/:
One way in which these two examples differ is that the second is a harmonic
function (that is, it satisfies the Laplace equation), and the roots of harmonic

26 C. Bénéteau and N. Hudson
polynomials have been studied intensively in the last 15 years. Their rational
counterparts also arise naturally in the study of gravitational lenses. Let us first turn
to a discussion of what is known for harmonic polynomials.
2.2 Harmonic Polynomials
We begin with a definition.
Definition 2.1 A harmonic polynomial f.z/is a function that can be written as
f.z/Dp.z/Cq.z/; where pand qare complex (analytic) polynomials in z.
One can ask whether there is a version of the Fundamental Theorem of Algebra
that holds for these harmonic polynomials, that is, how many solutions can the
following equation have, where pand qare both (analytic) polynomials, of degree
nand m, respectively?
p.z/Dq.z/(1)
Notice that if nDm, then the number of solutions might be infinite, as previously
discussed, but one can show (see [24,25]) that if n>m, the number of solutions
is finite. Sheil-Small became interested in this question from the point of view
of harmonic mappings and questions related to the Bieberbach Conjecture, and
conjectured in 1992 [22]thatifn>m,Eq.(1) has at most n2solutions. In his 1998
PhD thesis [24,25], Wilmshurst used Bézout’s Theorem to prove this conjecture.
Intuitively,one can reduce (1) to two systems of real polynomialequations of degree
n, and counting intersections of that system of curves gives n2:Wilmshurst showed
that n2is sharp by considering examples where mDn1. He then conjectured
that if 1m<n1; an even lower bound might occur, namely, that the largest
number of zeros of (1) in that case is 3n2Cm.m1/. In particular, for mD1,
this intriguing conjecture stated that the maximum number of zeros of an equation
of the type p.z/DNz;where pis an (analytic) polynomial of degree n>1, is equal
to 3n2. Several people (Bshouty, Crofoot, Lizzaik, Sarason, and others) worked
on this and related questions in harmonic mappings, and made partial progress (see,
e.g., [6]). Significant progress occurred on this problem in 2002, when Khavinson
and ´
Swia¸tek [15] proved Wilmshurst’s Conjecture for mD1. They used the fact that
fixed points of p.z/are also fixed points of the analytic polynomial Q.z/Dp.p.z//
(of degree n2), and then used complex dynamics and the argument principle for
harmonic functions to count the different types of fixed points that can occur. They
noticed that the bound is sharp by considering quadratic polynomials (nD2).
Geyer [9] later showed this bound to be sharp for all n:In 2016, Khavinson et al.
[16] employed a clever method of comparing Minkowski areas of Newton polygons
generated by harmonic polynomials to refine known results about maximal numbers
of zeros of (1) when the degree mof qsatisfies 1<m<n1; where nis the
degree of p. For example, they show that for each n>m;there exists a harmonic
polynomial h.z/Dp.z/Cq.z/with at least m2CmCnroots.

A Survey on the Maximal Number of Solutions of Equations Related to... 27
Instead of considering maximal numbers of roots of harmonic polynomials,
one can also think of probabilistic interpretations, that is: how many roots does
such an equation have on average? In 2009, Li and Wei analyzed the probabilistic
distribution of random harmonic polynomials [19]. This resulted in an exact
equation for the expected number of roots of (1), a rather complicated Lebesgue
integral. Li and Wei used this formula to show that, when the degree nof pis
equal to the degree mof q, the expected number behaves like 
4n3=2 as n!1;
while if mD’nCo.n/with 0’<1;the expected number is asymptotically
nas n!1:Others began trying to come up with examples, either to prove or
disprove Wilmshurst’s Conjecture for m2. In 2013, Lee et al. [17] used methods
of algebraic geometry to conclude that for mDn3and n4, there exists
harmonic polynomials pnand qmfor which the number of roots exceeds
n24nC4$n2
arctan pn22n
n%C2;
which yields an infinite number of counterexamples to Wilmshurst’s conjecture in
general. Hauenstein, Lerario, Lundberg, and Mehta generalized these counterex-
amples in 2014 [10] by using an experimental, certified-counting approach. In
particular, they conjectured that for all even n, there exist polynomials pwith degree
nand qwith degree n=2 such that Eq. (1) has exactly n2=2 nC12 roots. In
2016 [18], Lerario and Lundberg confirmed and sharpened the results of Li and
Wei that had previously been obtained by computer experiments. Thomack [23]
recently showed, using a slightly different probabilistic model, that if mis fixed, the
expectation of the number of zeros is asymptotically nas n!1:It is worth noting
in addition that by the Argument Principle, if phas degree n>1;then the number
of zeros of p.z/DNzis at least n:
In spite of all of this progress, the following basic open question remains.
Open Question 2.1 If pis a polynomial of degree n>3and qis a polynomial of
degree mwith 1<m<n1, what is the maximal number of roots that the equation
p.z/Cq.z/D0can have?
Any progress on this question would be of interest, in particular, showing whether
or not the maximal number of roots is linear in nfor mfixedwouldbealargestep
forward.
2.3 Rational Functions and Gravitational Lensing
In 2006, Khavinson and Neumann (see [13]) had the idea to replace the polynomial
pin the first simplest harmonic polynomial equation p.z/DNzby a rational function
r.z/WD p.z/
q.z/of degree n,wherepand qare polynomials, and by definition nD
maxfdeg p;deg qg:That is, they investigated the maximal number of solutions of
the equation
r.z/DNz:(2)

28 C. Bénéteau and N. Hudson
They proved that the maximal number of solutions to (2)is5n5. In the process
of working on this problem, they discovered an amazing connection with a problem
in mathematical physics related to gravitational lensing. In fact, the astrophysicist
Rhie [20] had conjectured this upper bound and had a beautiful simple geometric
construction that showed that this bound can actually occur for all n>1;thus
confirming that the result is sharp. In [5], the authors studied intermediate numbers
of solutions of Eq.(2), and showed that any number of simple zeros allowed by the
Argument Principle occurs.
Gravitational lensing is the bending of light by a gravitational field. This
phenomenon was predicted by general relativity. In fact, Einstein’s theory was first
proven by Sir Arthur Eddington via observation of a lensing event during a total
solar eclipse in 1919. One way of formulating the gravitational lensing problem
involves using multiple complex planes as the stage for the physical configuration
of objects involved—a light-emitting source, a massive object (the lens), and the
observer. More specifically, to mathematically model the lens, we consider three
parallel planes (Fig. 2): the observation plane where the observer is located, the lens
plane containing the galaxy (or other massive object) acting as the lens, and, located
on the opposite side of the lens plane from the observer,the source plane where the
light source is positioned.
The lens equation takes the following form:
wDzZ

d.—/
NzN
—”Nz;(3)
Source plane
L
S1S2
S
Lens plane
Fig. 2 Simple diagram of the source and lens planes

A Survey on the Maximal Number of Solutions of Equations Related to... 29
where wis the position of the light source, zis the position of a lensed image, ”is
a gravitational shearing term, dis a measure related to the gravitational masses,
and is the domain containing the support of . The shearing term accounts for
the effects of an external gravitational field on the lens system, such as the pull of a
nearby galaxy. Many astrophysicists(Witt, Mao, Peters, Rhie, Burke, among others)
have worked on estimates of numbers of solutions of the lens equation and related
problems. For a nice discussion of the history of the discovery of the lensing effect
and on early estimates from astrophysicists on number of images that can occur, see
for example [8,14] and the references therein.
The measure dcan be discrete (representing point masses), or a continuous
distribution of mass on a compact set. Neither model is completely physically
accurate: galaxies are not actually continuous objects, and stars are not point masses,
but this model is accurate enough to predict known physical phenomena, as seen in
some of the pictures below. The discrete form of the lens equation produced by a
discrete measure at npoint masses becomes the following:
wDz
n
X
jD1
¢j
NzNzj”Nz(4)
where w,z,and”play the same roles as before, and ¢jis a coefficient related to the
mass and the so-called “optical depth” of each point mass on the lens plane.
When a gravitational lens, a light source, and an observer are colinear, the lensed
image does not appear as a point, but a circle or ellipse! The halo of light that
is formed is called an ‘Einstein Ring’ (see Fig. 3). Interestingly, this result can be
obtained from either the discrete configuration, or the continuous distribution of an
(a) (b)
Fig. 3 Images of Einstein rings captured by the Hubble space telescope. (a) Light from a
blue galaxy distorted by a red galaxy. Credit: NASA, ESA. Source: https://www.nasa.gov/sites/
default/files/thumbnails/image/15861603283_3579db3fc6_o.jpg.(b) An elliptical galaxy which
is lensing light from two different background galaxies. Credit: NASA, ESA, R. Gavazzi and
T. Treu (University of California, Santa Barbara). Source: https://www.nasa.gov/images/content/
207624main_double_einstein_full.jpg

30 C. Bénéteau and N. Hudson
elliptical (or circular) galaxy. In the discrete case, simply set w,”,andNz1equal to
zero, and what follows immediately is the complex equation of a circle with radius
p¢1. The proof that in the continuous case, you obtain only a circle or an ellipse
is more difficult and can be found in [8]. Again, although maximal numbers of
solutions of Eq. (2) are known, as soon as the right hand side has degree larger than
1, the question regarding the exact number of images remains open.
3 Gravitational Lensing: Continuous Mass Distributions
Several authors have studied the question of number of roots of the lens equation
for continuous densities in different contexts (see [2–4,8]). Let us first examine the
case of ellipses with uniform mass densities.
3.1 Ellipse with Uniform Mass Density
Fassnacht et al. [8] explored the case in which the gravitational lens was an ellipse
with uniform mass density. In what follows, we describe their method and results.
Conjugating lens equation (3) and using the fact that d.—/ D1
dA.—/; a uniform
mass density, we rewrite the lens equation in the form:
NwDNz1
Z

dA.—/
z—”z
where WD fzDxCP{yWx2
a2Cy2
b2<1g,anddA WD dA.—/ Dd—dN
—
2iis area measure.
One can use the complex form of Green’s Theorem to rewrite the integral in the
lens equation for the ellipse as a line integral. The result will vary depending on
whether z2or z…. In this context, zbeing inside or outside of corresponds
to the observed image appearing inside the galaxy (a “dim” image) or outside (a
“bright” image).
For z…:
1
Z

dA
z—D 1
2i“

d—dN
—
z—D1
2P{Z
@
N
—d—
—z:
For z2:
1
Z

dA
z—D 1
2i“

d—dN
—
z—DNzC1
2P{Z
@
N
—d—
—z:

A Survey on the Maximal Number of Solutions of Equations Related to... 31
Both cases involve the same Cauchy integral:
1
2P{Z
@
N
—d—
—z:
In order to rewrite this integral, Fassnacht et al. [8] used the Schwarz function for
the ellipse to express N
—as an analytic function along the boundary. Indeed, suppose C
is an analytic arc given by f.x;y/D0: Rewriting this equation in complex notation
gives
fzCNz
2;zNz
2P{g.z;Nz/D0:
If rfD@g
@Nz¤0along C, then there exists a function S.z/; analytic in a
neighborhood of Csuch that S.z/DNzon C. Such a function S.z/is called the
Schwarz function of C. For more detail on the Schwarz function, see [7,21].
After substituting the Schwarz function of the ellipse into the integral, applying
Cauchy’s formula, and performing some algebraic manipulations, Fassnacht et al.
[8] transformed the lens equation into a polynomial in zand Nz. Rewriting this lens
equation as a system of two real equations, they derived that the maximum number
of images produced by an elliptical lens is four outside and one inside.
3.2 Investigating the Limaçon
Let us now use this same method to investigate the number of images formed by a
limaçon lens.
3.2.1 The Lens Equation
The limaçon (Fig. 4) can be thought of as the shape whose equation in polar
coordinates is given by: rDaCbcos ™; where aand bare real constants.
The most familiar example is the cardioid (the dotted shape in Fig. 4), which
occurs when jajDjbj. (In some cases, the term limaçon is used interchangeably
with cardioid, but in this paper, we will call the more general shape a limaçon.) It
is easy to see that the area of a limaçon with no internal “loops” (i.e., jajjbj)is
.a2Cb2
2/.
Let WD f zDreP{™ 2Cjr<aCbcos ™g,wherea;bare fixed. For simplicity,
let us consider aand bpositive. Let D@ be the limaçon. The lens equation for
a limaçon-shaped lens with uniform mass density d.—/ D2
3 dA.—/; is:
NwDNz2
3 Z

dA.—/
z—”z

32 C. Bénéteau and N. Hudson
4
–8 –6 –4 –2 0 2 4 6
2
–2
–4
r = 2 – 6 cos(θ)
r = 3 + 3 cos(θ)
r = 5 + 2 cos(θ)
Fig. 4 Graph of three different limaçons
where dA WD dA.—/ Dd—dN
—
2i, and the integral includes a normalizing factor so that
the area of a limaçon with aDbD1is 1. Using the complex form of Green’s
Theorem to rewrite the limaçon lens equation as a line integral, as in [8], we notice
that for z…:
2
3 Z

dA
z—D 1
3i“

d—dN
—
z—D2
3
1
2P{Z
@
N
—d—
—z;
while for z2:
2
3 Z

dA
z—D 1
3i“

d—dN
—
z—D2
32
4NzC1
2P{Z
@
N
—d—
—z3
5:
Note that this differs from the elliptical case only by the normalizing factor of 2
3.
3.2.2 The Schwarz Function
Let us now revisit the Schwarz function of the limaçon (see also [7]). Squaring the
equation rDaCbcos ™so that r2Dar Cbr cos ™and using complex coordinates

A Survey on the Maximal Number of Solutions of Equations Related to... 33
.—; N
—/ gives —N
—Da.—N
—/ 1
2Cb
2.— CN
—/: Performing some algebraic manipulation
leads to
—N
—b
2.— CN
—/ Da.—N
—/ 1
2
.—N
—/2Cb2
4.—2CN
—2C2—N
—/ b.—2N
—C—N
—2/Da2—N
—
N
—2—b
22
N
—— b—b
2Ca2Cb2
4—2D0:
Since this equation is quadratic in N
—, we can apply the quadratic formula to obtain
N
—D1
2.— b
2/2—b—b
2Ca2
˙1
2.— b
2/2s—2b—b
2Ca22
4—b
22b2
4—2
D—
2.— b
2/2"b—b
2Ca2˙as2b—Ca2b2
2b#:
To simplify notation, let us define AWD a2b2
2b;and notice that for —2, the positive
choice of sign in the above expression is necessary: that is, for —2; we have
N
—D®1.—/ C®2.—/;
where ®1.—/ D—
2.—b
2/2b—b
2Ca2and ®2.—/ Da—p2b.—CA/
2.—b
2/2:Let us now further
assume that the constants aand bsatisfy a>b>0;and make a choice of branch
cut along the ray .1;A. With that choice made, both ®1and ®2are analytic in
a neighborhood of ; and thus the Schwarz function S.—/ D®1.—/ C®2.—/: Note
that ®1is analytic in Cfb=2g,and®2is analytic in fb=2g,andthatb=2 lies
inside .
3.2.3 Solving the Lens Equation and Counting Solutions
Recall that in order to find the lens equation for either zinside or outside the domain
, we need to calculate the integral
Z
@
N
—d—
—z;

34 C. Bénéteau and N. Hudson
which can be rewritten using the Schwarz function from the previous section as
Z
@
®1.—/ C®2.—/
—zd—:
Now for z…,then ®1.—/C®2.—/
—zis analytic inside except at the pole b=2,and
therefore by the Residue Theorem applied to the integrand as a function of —(with z
fixed),
1
2P{Z
@
®1.—/ C®2.—/
—zd—DRes ®1.—/ C®2.—/
—zIb=2:
A somewhat lengthy but straightforward calculation gives that
Res ®1.—/ C®2.—/
—zIb=2D2a2zCb2.b=2 z/
2.b=2 z/2:
On the other hand, if z2, then the integrand has residues both at b=2 as well
as at z, and thus, we need to add the residue at zto the previous expression, and we
get that:
1
2P{Z
@
®1.—/ C®2.—/
—zd—Db.b=2 z/.bz/a2zCazp2b.zCA/
2.zb=2/2:
That is, we have proved the following.
Theorem 3.1 Suppose WD f zDreP{™ 2Cjr<aCbcos ™gis the interior
of a limaçon with constants chosen so that a >b>0;with constant mass density
d.—/ D2
3 dA.—/: Then for z …, the lens equation is
NwDNzC2
32a2zCb2.b=2 z/
2.b=2 z/2”z;(5)
while for z 2, the lens equation is:
NwD1
3NzC2
3"b.b=2 z/.bz/a2zCazp2b.zCA/
2.zb=2/2#”z:(6)
Now we can represent zand Nzin terms of real variables xand y, and letting NwDsP{t,
where sand tare real constants, we can use these lens equations to count solutions.
In particular, notice that Lens Equation (5) becomes a system of two real equations
of degree 3, and therefore, by Bézout’s Theorem, there are at most 9 images outside
the limaçon. Notice that theorem of Khavinson and Neumann for rational functions
gives 5n5for nD3in this case, thus 10 solutions, so this approach gives a slight
improvement in that case. On the other hand, if there is no shearing term (”D0),
then nD2, and therefore the Khavinson-Neumann theorem gives 5solutions, while
Bézout’s theorem still gives 9.

A Survey on the Maximal Number of Solutions of Equations Related to... 35
In the case of the Lens Equation (6), the Khavinson-Neumann theorem does not
apply at all, since the equation does not involve a rational function. Rewriting the
equation using real variables, squaring both sides, and canceling common factors of
.zb=2/ gives rise to a system of two real equations. The equation corresponding
to the real part of the lens equation is of degree 4, regardless of the choice of ”.
However, the imaginary part is of degree 3 if ”D0and of degree 4 otherwise.
Hence, by Bézout’s theorem, there are at most 16 solutions inside the limaçon if the
shearing term doesn’t vanish and at most 12 if it does. Thus, we have the following.
Theorem 3.2 Suppose WD f zDreP{™ 2Cjr<aCbcos ™gis the interior
of a limaçon with constants chosen so that a >b>0;with constant mass density
d.—/ D2
3 dA.—/: Then there are at most 9 solutions to the lens equation (5)if the
shearing term ”¤0, and at most 5 solutions if ”D0; while there are at most
16 solutions to the lens equation (6)if the shearing term ”¤0, and at most 12
solutions if ”D0:
It is clear that these are significant overestimates, especially for the images inside
the limaçon, since information seems to be lost when squaring and moving to a
generic estimate of real variable solutions. However, having a concrete expression
for the lens equation depending on the parameters of the limaçon may lead to
numerical experiments that will give rise to the appropriate conjecture for the
number of images. Indeed, so far, numerical experiments show only 4 images inside
with a shearing term present and 1 outside (see Figs.5and 6).Basedonthese
numerical experiments, we state the following conjecture.
40
30
20
10
–10
–20
–30
–40
–40 –30 –20 –10 10 20 30 40
0
Real part of z inside
Limacon
Imaginary part of z inside
y
~
t
~
Position of light source
Fig. 5 Zero set of the system of equations resulting from the lens equation for the limaçon given
by rD15 C6cos ™, a source position wD5.1 CP{/, and a shearing term of ”D2:5.Thereare4
images, located at the crossings of the solid and dashed curves that lie within the limaçon

36 C. Bénéteau and N. Hudson
30
20
10
–10
–20
–30
–30 –20 –10 10 20 300
y
~
x
~
Position of light source
Limacon
Real part of z outside
Imaginary part of z outside
Fig. 6 Zero set of the system of equations resulting from the lens equation for the limaçon given
by rD11 C5:5 cos ™, a source position wD5.1 C3P{/, and a shearing term of ”D0.Thereis1
image, located at the crossing of the dotted and dashed curves that lies outside of the limaçon
Conjecture 3.3 Suppose WD f zDreP{™ 2Cjr<aCbcos ™gis the interior
of a limaçon with constants chosen so that a>b>0;with constant mass density
d.—/ D2
3 dA.—/: Then there is at most 1 solution to the lens equation (5)andthere
are at most 4 solutions to the lens equation (6).
One might find this conjecture plausible due to the fact that we can think of the
limaçon as a perturbation of the ellipse, and thus, the same estimates on the maximal
number of zeros (see [8]) is reasonable.
Finally, the limaçon is an example of a quadrature domain, and it is known
that for these domains, the lens equation for solutions inside the shape (the “dim”
images) is of the form a.z/DNzfor an algebraic function a(seeRemark2onp.14of
[8]). Thus, finding maximal numbers of solutions for such shapes, or equivalently,
for lens equations involving algebraic functions seems like a challenging problem.
3.3 Lens Equations Involving Transcendental Functions
Certain choices of mass distributions lead to lens equations involving transcendental
functions. In [8], the authors discuss the situation of a more physically relevant
density than the constant mass distribution, that is “isothermal” densities, which
are densities that are constant on homothetic ellipses (rather than confocal). These

A Survey on the Maximal Number of Solutions of Equations Related to... 37
densities are obtained by projecting the more realistic 3-dimensional density 1=r2
that is proportional to the square of the distance rfrom the origin. In this 3-
dimensional context, the gas in the galaxy has constant temperature, hence the term
“isothermal” (see, e.g., [11]). For the ellipse, such a choice results in a lens equation
involving a branch of arcsin, a transcendental function! (Note that in the case of a
circle, the situation is much simpler and gives rise to what is called a Chang-Refsdal
lens, see [1].) In 2010, Khavinson and Lundberg [12] investigated the transcendental
equation
arcsin k
NzCNwDz;(7)
which is the lens equation for the ellipse with isothermal density. They showed that
an upper bound on the number of images outside the ellipse (the so-called “bright”
images) is at most 8. In that same year, Bergweiler and Eremenko [3] showed that
the sharp bound is actually 6, and indeed any number of solutions from 1 to 6 can
occur.
In [2], the authors considered isothermal densities with an added twist factor, so
that the mass density has a spiral structure controlled by a real parameter s.More
specifically, they considered the density d.—/ D¡.j—j/dA.—/ where ¡.r/DM=r;
rDe™=s;with some restrictions on s, and here rDj—jand ™are polar coordinates.
They calculated the lens equation in this case and showed that it involves the Gauss
hypergeometric function (see Theorem 2.1 of [2]).
In general, very little is known about maximal numbers of solutions of equations
involving transcendental functions. A recent result of Bergweiler and Eremenko
[4] gives upper and lower bounds on the number of solutions Nof the equation
p.z/log jzjCq.z/D0; where pand qare (co-prime) polynomials of degree n
and mrespectively: they show that maxfn;mgN3maxfn;mgC2n;and that
the estimate is sharp for many values of nand m. It is clear that investigating more
situations stemming from a rich variety of transcendentalfunctions will prove useful
and important for the further development of the subject.
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