# Some equations involving the Gamma function

Sebastian Eterović  School of Mathematics, University of Leeds, Leeds LS2 9JT,
UK<EMAIL_ADDRESS>and  Adele Padgett  Department of Mathematics and
Statistics, McMaster University, Hamilton, ON, Canada<EMAIL_ADDRESS>
(Date: October 2, 2023)

###### Abstract.

Let $V\subseteq\operatorname{\mathbb{C}}^{2n}$ be an algebraic variety with no
constant coordinates and with a dominant projection onto the first $n$
coordinates. We show that the intersection of $V$ with the graph of the
$\Gamma$ function is Zariski dense in $V$.

###### Key words and phrases:

Gamma function, existential closedness, Rouché’s theorem

###### 2020 Mathematics Subject Classification:

30C15, 33B15, 30D35, 32A60

SE was supported by EPSRC fellowship EP/T018461/1.

##  1\. Introduction

In this paper we prove the following result.

######  Theorem 1.1.

Let $n$ be a positive integer and let
$V\subseteq\operatorname{\mathbb{C}}^{2n}$ be an algebraic variety with no
constant coordinates. Let $\pi_{1}:\mathbb{C}^{2n}\to\mathbb{C}^{n}$ denote
the projection onto the first $n$ coordinates. If $\dim\pi_{1}(V)=n$, then $V$
has a Zariski dense set of points of the form
$(\mathbf{z},\Gamma(\mathbf{z}))$.

This follows a long list of similar results for different functions that have
appeared in recent years, such as for the complex exponential function
([AKM23, BM17, DFT18, MM23, Mar06]), the exponential maps of abelian varieties
([AKM23, Gal23b]), the modular $j$ function ([EH21, Gal21]), and the
uniformization maps of Shimura varieties ([EZ21]).

The unifying theme behind all these results is that of understanding the
algebraic properties of the function at hand. To be more precise, given a
transcendental holomorphic function $f$, what we aim for is to find a minimal
set of geometric conditions an algebraic variety $V$ should satisfy to ensure
that the intersection between $V$ and the graph of $f$ is Zariski dense in
$V$. By “geometric conditions” we mean conditions that make sense on any
algebraically closed field of characteristic zero. This is a natural question
and it is closely related to the questions about value distributions of $f$
with moving targets, a classical problem in complex analysis. In particular,
Theorem 1.1 completely solves the problem for the case $n=1$.

This problem has received different names: it has been called _Zilber’s
Nullstellensatz_ and _exponential algebraic closedness_ in the case of complex
exponentiation, after the highly influential work of Zilber on
pseudoexponentiation ([Zil02, Zil05]), and more generally it has been named
the _existential closedness problem_ (EC).

Our main theorem solves some cases of the problem for the $\Gamma$ function.
The articles cited above provide different methods for finding points in the
intersection of an algebraic variety $V$ and the graph of the corresponding
function $f$. We remark that all the functions considered in these articles
(exponential maps and automorphic functions) are periodic, and in each case,
the methods exploit the periodicity of the function being studied. Moreover,
many of the methods provide not only the existence of points in the
intersection of $V$ with the graph of $f$, but also an indication of how these
points are distributed. Our aim is to develop an approach that works for the
$\Gamma$ function (a non-periodic function of great interest) while still
providing information about the distribution of intersection points. For this
we will extend some of the methods used in [EH21] and [EZ21] which rely on
Rouché’s theorem. We will also be aided by the Argument Principle, a close
companion to Rouché’s theorem. This will allow us not only to prove Theorem
1.1, but also to solve certain systems of analytic equations involving
$\Gamma$, see Proposition 4.2. For example, this will show that the equation
$\Gamma(z)=\exp\left(\frac{1}{z}\right)$ has infinitely many solutions.

We remark that this venture is motivated not only by solving this type of
problem for various kinds of functions, but also by the desire for an
underlying method that can be adapted to all cases. We strongly believe that
the methods presented here can be adapted to the case of exponential maps of
abelian and semi-abelian varieties, at least to obtain analogues of Theorem
1.1. Furthermore, Theorem 1.1 has the same level of generality as many of the
results of the articles cited at the beginning, except for [Gal23b, Gal21,
Gal23a, MM23] which have other kinds of restrictions.

The structure of the paper is as follows. In §2 we give some background on the
$\Gamma$ function, including some transcendence questions and a description of
the behavior of $\Gamma$ on the upper right quadrant of
$\operatorname{\mathbb{C}}$. Then in §3 we give two proofs of Theorem 1.1 in
the case $n=1$. Finally in §4 we prove the main result using some of the
methods in §3.

###  1.1. Notation

  1. (a)

We will denote tuples of complex numbers using bold face letters. So a tuple
$(z_{1},\ldots,z_{n})\in\operatorname{\mathbb{C}}^{n}$ will be denoted as
$\mathbf{z}$. Furthermore, if $f$ denotes a one-variable function, then we
write $f(\mathbf{z})$ to denote the tuple $(f(z_{1}),\ldots,f(z_{n}))$.

  2. (b)

We will use $\Gamma$ to denote both the function in one complex variable, and
also for its extension to $n$-variables
$\Gamma:\left(\operatorname{\mathbb{C}}\setminus\operatorname{\mathbb{Z}}^{\leq
0}\right)^{n}\to\operatorname{\mathbb{C}}^{n}$ via
$\Gamma(\mathbf{z})=(\Gamma(z_{1}),\ldots,\Gamma(z_{n}))$. Any ambiguities
will be avoided by context.

  3. (c)

We let $\Re(z)$ and $\Im(z)$ denote the real and imaginary parts of a complex
number $z\in\operatorname{\mathbb{C}}$, respectively.

  4. (d)

Given $z\in\operatorname{\mathbb{C}}$ and $\epsilon>0$, we let $B(z,\epsilon)$
denote the open ball centered at $z$ of radius $\epsilon$.

  5. (e)

When talking about the argument of a complex number
$z\in\operatorname{\mathbb{C}}$, we will always assume that the argument
function is defined on all of $\operatorname{\mathbb{C}}^{\times}$. Of course,
this implies that $\arg(z)$ is not continuous on all of
$\operatorname{\mathbb{C}}^{\times}$, but it is continuous when restricting to
the domain of a specific branch of logarithm. Unless otherwise stated, we
express the argument function as
$\arg:\operatorname{\mathbb{C}}^{\times}\to(-\pi,\pi]$ and if
$\Re(z),\Im(z)>0$, then $\arg(z)=\arctan\left(\frac{\Im(z)}{\Re(z)}\right)$.

  6. (f)

Given a subset $X\subset\operatorname{\mathbb{C}}^{n}$, we let $\overline{X}$
and $X^{\circ}$ denote the topological closure and interior of $X$,
respectively. We also use $\partial X$ to denote the boundary of $X$, that is,
$\partial X=\overline{X}\setminus X^{\circ}$.

### Acknowledgements

We would like to thank Vahagn Aslanyan, Gareth Jones, and Jonathan Kirby for
constructive discussions around the topics presented here.

##  2\. Background on $\Gamma$

We recall here some well-known results about $\Gamma$. For reference, see e.g.
[Rem98, Chapter 2]. The $\Gamma$ function is usually first defined on the
right half plane $\\{z\in\operatorname{\mathbb{C}}:\Re(z)>0\\}$ via the
following integral:

$\Gamma(z):=\int_{0}^{\infty}t^{z-1}\exp(-t)\mathrm{d}t.$

In particular we get that $\Gamma(1)=1$, $\Gamma(x)>0$ whenever $x>0$, and
$\overline{\Gamma(z)}=\Gamma(\overline{z})$ for every $z$ in the right half
plane. This function has an analytic continuation to
$\mathbb{C}\setminus\mathbb{Z}^{\leq 0}$ which can be described using the
following infinite product:

(2.1) $\Gamma(z)=\frac{\exp(-\gamma
z)}{z}\prod_{k=1}^{\infty}\left(1+\frac{z}{k}\right)^{-1}\exp\left(\frac{z}{k}\right),$

where $\gamma$ denotes the Euler-Mascheroni constant:111It is unknown whether
this number is irrational.

$\gamma=\lim_{n\rightarrow\infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\log(n)\right).$

$\Gamma$ can also be defined via the following limit

(2.2) $\Gamma(z)=\lim_{n\rightarrow\infty}\frac{n!n^{z}}{z(z+1)\cdots(z+n)}.$

Using (2.2), it is easy to see that whenever $\Re(z)>0$, we have that
$|\Gamma(z)|\leq\Gamma(\Re(z))$.

We note that $\Gamma$ is holomorphic on $\mathbb{C}\setminus\mathbb{Z}^{\leq
0}$ and meromorphic on $\mathbb{C}$, having a simple pole at every point of
$\mathbb{Z}^{\leq 0}$. It also satisfies the following difference equations:

  1. (a)

For every $z\in\mathbb{C}\setminus\mathbb{Z}^{\leq 0}$,

(2.3) $\Gamma(z+1)=z\Gamma(z).$

  2. (b)

For every $z\in\operatorname{\mathbb{C}}$,

(2.4) $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}.$

  3. (c)

For every
$z\in\operatorname{\mathbb{C}}\setminus\operatorname{\mathbb{R}}^{\leq 0}$ and
for every integer $n\geq 2$,

(2.5)
$\prod_{k=0}^{n}\Gamma\left(z+\frac{k}{n}\right)=(2\pi)^{\frac{1}{2}(n-1)}n^{\frac{1}{2}-nz}\Gamma(nz).$

In fact, a theorem of Wielandt shows that $\Gamma$ is the unique function
which is holomorphic on $\\{z\in\operatorname{\mathbb{C}}:\Re(z)>0\\}$, is
bounded on $\\{z\in\operatorname{\mathbb{C}}:1\leq\Re(z)<2\\}$, and satisfies
(2.3) and $\Gamma(1)=1$ (see [Rem98, Chapter 2, §2.4]).

We also recall Stirling’s formula: there is a holomorphic function $\mu(z)$ on
$\operatorname{\mathbb{C}}\setminus\operatorname{\mathbb{R}}^{\leq 0}$ such
that $\lim_{z\to\infty}\mu(z)=0$ and

(2.6) $\Gamma(z)=\sqrt{2\pi}z^{z-\frac{1}{2}}\exp(-z)\exp(\mu(z)).$

There are more precise descriptions of $\mu(z)$, see [Rem98, Chapter 2, §4.2].
In particular, when $\Re(z),\Im(z)>0$, there is $M>0$ such that if $|z|>M$
then $|\mu(z)|<1$, and so $\frac{1}{e}\leq|\exp(\mu(z))|\leq e$, where
$e=\exp(1)$. Thus, if we write $z=x+iy$ with $x,y>0$, we get that when $|z|>M$

(2.7)
$\frac{\sqrt{2\pi}}{e}|z|^{x-\frac{1}{2}}\exp\left(-y\arg(z)-x\right)\leq|\Gamma(z)|\leq
e\sqrt{2\pi}|z|^{x-\frac{1}{2}}\exp\left(-y\arg(z)-x\right).$

One can use (2.7) to obtain that $|\Gamma(x+iy)|$ tends exponentially to zero
as $|y|$ tends to $+\infty$. More precisely, if $K$ is a compact subset of
$\operatorname{\mathbb{R}}$, then there are $C>0$ and
$y_{0}\in\operatorname{\mathbb{R}}$ such that whenever $x\in K$ and
$|y|>|y_{0}|$ we have

(2.8) $|\Gamma(x+iy)|\leq
C\sqrt{2\pi}|y|^{x-\frac{1}{2}}\exp\left(-\frac{\pi}{2}|y|\right).$

###  2.1. Behavior of $\Gamma$ in the upper right quadrant

In this section, we always use $z$ to denote a complex number, $x$ its real
part, and $y$ its imaginary part. We denote the upper right quadrant as
$Q:=\\{z\in\mathbb{C}:\Re(z),\Im(z)>0\\}$. Let $\alpha$ denote the unique
positive real zero of $\Gamma^{\prime}$ ($\alpha\approx 1.4616\dots$, see
[Uch12]).

We first remark that $|\Gamma(z)|$ decreases exponentially to zero along
vertical lines (with $y$ tending to $+\infty$) by (2.8), and $|\Gamma(z)|$
grows exponentially along horizontal lines (with $x$ tending to $+\infty$) by
(2.7). For future reference, we recount this in the following lemmas and
corollaries (which are immediate).

######  Lemma 2.1.

For fixed $x>0$, $|\Gamma(x+iy)|$ decreases exponentially in $y$.

######  Lemma 2.2.

For fixed $y$ and $x\geq\alpha$, $|\Gamma(x+iy)|$ increases exponentially in
$x$.

######  Corollary 2.3.

Let $A(z)$ be any non-zero algebraic function. For fixed $x>0$, if $A(x+iy)$
is defined for all $y$ large enough, then
$\displaystyle\lim_{y\to+\infty}\frac{\Gamma(x+iy)}{A(x+iy)}=0$.

######  Corollary 2.4.

Let $A(z)$ be any non-zero algebraic function. For fixed $y$, if $A(x+iy)$ is
defined for all large enough $x$, then
$\displaystyle\lim_{x\to+\infty}\left|\frac{\Gamma(x+iy)}{A(x+iy)}\right|=\infty$.

Given positive real numbers $r_{1}$ and $r_{2}$, we define

$Q_{r_{1},r_{2}}:=\\{z\in Q:\Re(z)>r_{1},\Im(z)>r_{2}\\}.$

Now we will study the family of sets $C_{r}:=\\{z:|\Gamma(z)|=r\\}$ for
$r\in\operatorname{\mathbb{R}}^{+}$. We will focus on how these sets behave in
$Q_{\alpha,0}$, though analogous statements hold in its reflection over the
real axis because $\overline{\Gamma(z)}=\Gamma(\overline{z})$.

######  Proposition 2.5.

For each $r\in\operatorname{\mathbb{R}}^{+}$, the set of $x+iy\in
Q_{\alpha,0}\cap C_{r}$ form a single $C^{1}$ curve with positive slope and no
horizontal or vertical asymptotes.

###### Proof.

Let $r\in\operatorname{\mathbb{R}}^{+}$. By Corollaries 2.3 and 2.4 and the
intermediate value theorem, there are $x_{0}>\alpha$ and $y_{0}>0$ be such
that $|\Gamma(x_{0}+iy_{0})|=r$. By Lemma 2.1,
$\displaystyle\frac{\partial|\Gamma(x+iy)|}{\partial y}(x_{0}+iy_{0})\neq 0$,
so we can use the implicit function theorem to obtain a unique $C^{1}$
function $y(x)$ on some interval $(x_{0}-\epsilon,x_{0}+\epsilon)$ such that
$C_{r}$ matches the graph of $y(x)$ in a neighborhood of $(x_{0},y_{0})$.
Again using the intermediate value theorem and the fact that $|\Gamma(x+iy)|$
strictly decreases vertically and strictly increases horizontally, the slope
of the implicit function $y(x)$ must be positive.

Now since $|\Gamma(x+iy)|$ is continuous and
$\displaystyle\frac{\partial|\Gamma(x+iy)|}{\partial y}$ is nonzero in the
upper right quadrant, we can extend the implicit function $y(x)$ on the left
all the way to either $x=\alpha$ or the $x$-axis so that it is still a $C^{1}$
curve with positive slope satisfying $|\Gamma(x+iy(x))|=r$. On the right, we
can also extend $y(x)$, and we will show that the extension does not approach
a vertical or horizontal asymptote. If $y(x)$ has a vertical asymptote at
$x=a$, then $|\Gamma(a+1+iy)|>r$ for all $y>0$ by Lemma 2.2, but that
contradicts Corollary 2.3. If $y(x)$ has a horizontal asymptote at $y=b$, then
$|\Gamma(x+i(b+1))|<r$ for all $x>0$ by Lemma 2.1, but that contradicts
Corollary 2.4.

Let $K_{r},K_{r}^{\prime}\subset Q_{\alpha,0}\cap C_{r}$ denote two
irreducible curves. $K_{r}$ and $K_{r}^{\prime}$ must be identically equal if
they intersect because otherwise the set of $(x,y)$ such that
$|\Gamma(x+iy)|=r$ would not locally be the graph of a function. If $K_{r}$
and $K_{r}^{\prime}$ are disjoint, let $y_{*}$ be least such that there are
$x,x^{\prime}$ with $(x,y_{*})\in K_{r}$ and $(x^{\prime},y_{*})\in
K_{r}^{\prime}$. Then $x\neq x^{\prime}$ but
$|\Gamma(x+iy_{*})|=r=|\Gamma(x^{\prime}+iy_{*})|$, which contradicts Lemma
2.2. ∎

######  Lemma 2.6.

Given $r\in\operatorname{\mathbb{R}}^{+}$, let $y_{r}(x)$ denote the function
whose graph is $C_{r}\subset Q_{\alpha,0}$. For every $M_{1}>0$ there is
$M_{2}>0$ such that for all $r\in\operatorname{\mathbb{R}}^{+}$ we have
$x>M_{2}\implies y_{r}^{\prime}(x)>M_{1}$. Furthermore $y^{\prime}_{r}(x)\geq
2\log\left(\lfloor x\rfloor\right)-2$. In particular,
$\displaystyle\lim_{x\to+\infty}y_{r}^{\prime}(x)=+\infty$.

###### Proof.

Since $|\Gamma(x+iy_{r}(x))|=r$, $\frac{d}{dx}(|\Gamma(x+iy_{r}(x))|)=0$,
i.e.,

$\displaystyle 0$ $\displaystyle=\frac{d}{dx}\left(\frac{e^{-\gamma
x}}{\sqrt{x^{2}+y_{r}^{2}}}\prod_{n=1}^{\infty}\frac{e^{x/n}}{\sqrt{\left(1+\frac{x}{n}\right)^{2}+\left(\frac{y_{r}}{n}\right)^{2}}}\right)$
$\displaystyle=r\left(-\gamma-\frac{y_{r}y_{r}^{\prime}-x}{x^{2}+y_{r}^{2}}+\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{y_{r}y_{r}^{\prime}+n+x}{(n+x)^{2}+y_{r}^{2}}\right)\right).$

Solving for $y_{r}^{\prime}$ gives

$y^{\prime}_{r}=\frac{-\gamma-\frac{x}{x^{2}+y_{r}^{2}}+\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{n+x}{(n+x)^{2}+y_{r}^{2}}\right)}{\frac{y_{r}}{x^{2}+y_{r}^{2}}+\sum_{n=1}^{\infty}\frac{y_{r}}{(n+x)^{2}+y_{r}^{2}}}.$

We remark that since $x\geq 1$, then
$\frac{x}{x^{2}+y_{r}^{2}}\leq\frac{1}{x}\leq 1$, and as $y_{r}\geq 0$, then
$\frac{y_{r}}{x^{2}+y_{r}^{2}}\leq\frac{y_{r}}{1+y_{r}^{2}}\leq\frac{1}{2}$.
Also, the sum $\sum_{n=1}^{\infty}\frac{y_{r}}{(n+x)^{2}+y_{r}^{2}}$ is
bounded above by
$y_{r}\frac{\pi^{2}}{6}=y_{r}\sum_{n=1}^{\infty}\frac{1}{n^{2}}$. So

$y^{\prime}_{r}\geq\frac{-\gamma-1+\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{n+x}{(n+x)^{2}+y_{r}^{2}}\right)}{\frac{1}{2}+y_{r}\frac{\pi^{2}}{6}}\geq-2\gamma-2+2\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{n+x}{(n+x)^{2}+y_{r}^{2}}\right).$

The sum
$\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{n+x}{(n+x)^{2}+y_{r}^{2}}\right)$
gets arbitrarily large as $x\to+\infty$:

$\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{n+x}{(n+x)^{2}+y_{r}^{2}}\right)$
$\displaystyle\geq\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+x}\right)$
$\displaystyle\geq\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+\lfloor
x\rfloor}\right)$ $\displaystyle=\sum_{n=1}^{\lfloor x\rfloor}\frac{1}{n}$
$\displaystyle\geq\log\left(\lfloor x\rfloor\right)+\gamma.$

Altogether,

$\displaystyle y_{r}^{\prime}$
$\displaystyle\geq-2\gamma-2+2\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{n+x}{(n+x)^{2}+y_{r}^{2}}\right)$
$\displaystyle\geq-2\gamma-2+2\left(\log\left(\lfloor
x\rfloor\right)+\gamma\right)$ $\displaystyle=2\log\left(\lfloor
x\rfloor\right)-2$

Since this last expression no longer depends on $r$, this proves the result. ∎

Next, we show that along each curve $C_{r}$ in $Q_{\alpha,0}$,
$\Gamma(x+iy(x))$ cycles around the circle of modulus $r$ faster as $x$ gets
larger.

######  Lemma 2.7.

Let $r\in\operatorname{\mathbb{R}}^{+}$ and let $y_{r}=y_{r}(x)$ be the
function whose graph is $C_{r}\subset Q_{\alpha,0}$. For every $M_{1}>0$ there
is $M_{2}>0$ such that for all $r\in\operatorname{\mathbb{R}}^{+}$ we have

$x>M_{2}\implies\frac{d}{dx}\left(\arg\Gamma(x+iy_{r}(x))\right)>M_{1}.$

In particular,

$\lim_{x\to+\infty}\frac{d}{dx}\left(\arg\Gamma(x+iy_{r}(x))\right)=+\infty.$

###### Proof.

Since $\log\Gamma(z)=\log|\Gamma(z)|+i\arg\Gamma(z)$, we get the following
expression for $\arg\Gamma(z)$ along $C_{r}$:

$\arg\Gamma(x+iy_{r})=-\gamma
y_{r}-\arctan\left(\frac{y_{r}}{x}\right)+\sum_{n=1}^{\infty}\left(\frac{y_{r}}{n}-\arctan\left(\frac{y_{r}}{n+x}\right)\right).$

Differentiating with respect to $x$ and writing $y_{r}^{\prime}$ to denote
$y_{r}^{\prime}(x)$, we get

$\displaystyle\frac{d}{dx}(\arg\Gamma(x+iy_{r}))=$ $\displaystyle-\gamma
y_{r}^{\prime}-\frac{xy_{r}^{\prime}-y_{r}}{x^{2}+y_{r}^{2}}+\sum_{n=1}^{\infty}\left(\frac{y_{r}^{\prime}}{n}-\frac{(n+x)y_{r}^{\prime}-y_{r}}{(n+x)^{2}+y_{r}^{2}}\right)$
$\displaystyle=$ $\displaystyle
y_{r}^{\prime}\left(-\gamma-\frac{x}{x^{2}+y_{r}^{2}}+\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{n+x}{(n+x)^{2}+y_{r}^{2}}\right)\right)$
$\displaystyle+\frac{y_{r}}{x^{2}+y_{r}^{2}}+\sum_{n=1}^{\infty}\frac{y_{r}}{(n+x)^{2}+y_{r}^{2}}$
$\displaystyle\geq$ $\displaystyle(2\log(\lfloor
x\rfloor)-2)\left(-\gamma-1+\log(\lfloor x\rfloor)+\gamma\right)$
$\displaystyle=$ $\displaystyle 2(\log(\lfloor x\rfloor)-1)^{2}$

Since this last expression no longer depends on $r$, this proves the result. ∎

######  Notation 2.8.

For each $z\in Q_{\alpha,0}$, define $z^{\ast}$ to be the the number of least
modulus greater than $|z|$ in $Q_{\alpha,0}$ such that
$\Gamma(z^{\ast})=\Gamma(z)$.

######  Corollary 2.9.

For all $z\in Q_{\alpha,0}$, we have
$\displaystyle|z-z^{\ast}|<\frac{\pi}{(\log(\lfloor\Re(z)\rfloor)-1)^{2}}$.

###### Proof.

Let $r=|\Gamma(z)|$ and let $y_{r}$ be the function whose graph is
$C_{r}\subset Q_{\alpha,0}$. Then $z^{\ast}$ is the first point in $C_{r}$
with $\Re(z^{\ast})>\Re(z)$ and $\arg\Gamma(z^{\ast})=\arg\Gamma(z)$. Note
that $|z-z^{\ast}|$ is less than the length of the segment of $C_{r}$ between
$z$ and $z^{\ast}$. By Lemma 2.7,
$\displaystyle\frac{d}{dx}\left(\arg\Gamma(x+iy_{r}(x))\right)>2(\log(\lfloor
x\rfloor)-1)^{2}$. So by the mean value theorem,

$|z-z^{\ast}|\leq\frac{2\pi}{2(\log(\lfloor
x\rfloor)-1)^{2}}=\frac{\pi}{(\log(\lfloor x\rfloor)-1)^{2}}\qed$

######  Remark 2.10.

We will use the bound found in Corollary 2.9 for large $\Re(z)$. The bound can
be improved for small values of $|z|$. For example, if $\Re(z),\Im(z)\geq 3$,
then it can be shown that $|z-z^{\ast}|<2\pi$.

Next we study the family of sets $\\{z\in
Q_{\alpha,0}:\arg\Gamma(z)=\theta\\}$ for fixed $\theta\in(-\pi,\pi]$.

######  Proposition 2.11.

For each $\theta\in(-\pi,\pi]$, the set of $x+iy\in Q_{\alpha,0}$ such that
$\arg\Gamma(x+iy)=\theta$ is a collection of disjoint $C^{1}$ curves whose
slopes are negative and approach zero as $x\to+\infty$.

###### Proof.

Let $\theta\in(-\pi,\pi]$. Let $z\in Q_{\alpha,0}$ and let $r=|\Gamma(z)|$. By
Corollary 2.9, there is a point $z_{0}$ in the segment of $C_{r}$ between $z$
and $z^{\ast}$ such that $\arg\Gamma(z_{0})=\theta$. Let $x_{0}=\Re(z_{0})$
and $y_{0}=\Im(z_{0})$. Since curves of fixed argument and modulus in
$\operatorname{\mathbb{C}}$ intersect at right angles, and since $\Gamma$ is
locally invertible on $Q_{\alpha,0}$ (see [Uch12]) and conformal, there must
be a neighborhood $U\ni x_{0}+iy_{0}$ such that
$U\cap\\{x+iy:\arg\Gamma(x+iy)=\theta\\}$ is a curve $K_{\theta}$ that
intersects $C_{r}$ at $x_{0}+iy_{0}$ at a right angle. Since the slope of
$C_{r}$ at $x_{0}+iy_{0}$ is positive (by Lemma 2.6), $K_{\theta}$ must pass
through $x_{0}+iy_{0}$ with negative slope. Specifically, if $y_{r}(x)$ is the
function whose graph is $C_{r}$, then the slope of $K_{\theta}$ at
$x_{0}+iy_{0}$ is $\frac{-1}{y_{r}^{\prime}(x)}$. Thus
$\frac{\partial\arg\Gamma(x+iy)}{\partial y}(x_{0}+iy_{0})\neq 0$, and we can
apply the implicit function theorem to obtain $K_{\theta}$ as the graph of a
unique $C^{1}$ function $y_{\theta}(x)$ with
$y_{\theta}^{\prime}(x)=\frac{-1}{y_{r}^{\prime}(x)}<0$.

We first show that we can extend $y_{\theta}(x)$ on the left to the line
$x=\alpha$. Applying the above argument at every $x+iy$ in $Q_{\alpha,0}$
shows that $\frac{\partial\arg\Gamma(x+iy)}{\partial y}$ is never 0. Suppose
$y_{\theta}(x)$ has a vertical asymptote at $x=a<x_{0}$. By conformality, that
would mean the slopes of $C_{r}$ curves in a neighborhood of $x=a$ would get
arbitrarily small as $y$ gets arbitrarily large. However, by Lemma 2.6,
$y_{r}^{\prime}(x)\geq 2\log(\lfloor x\rfloor)-2$, contradicting that
$y^{\prime}_{r}$ can get arbitrarily small.

The only barrier to extending $y_{\theta}(x)$ on the right to the whole
interval $(x_{0},+\infty)$ would be if $y_{\theta}(x)$ intersects the real
axis for some value of $x$. But this cannot happen because the positive reals
form a curve of fixed argument for $\Gamma$ (since $\Gamma$ is real valued and
positive on $(x_{0},+\infty)$) and different fixed argument curves cannot
intersect, since neither $\frac{\partial\arg\Gamma(x+iy)}{\partial y}$ nor
$\Gamma(x+iy)$ vanish in
$Q_{\alpha,0}\cup\\{x\in\operatorname{\mathbb{R}}:x>\alpha\\}$. ∎

######  Corollary 2.12.

Given $\theta\in(-\pi,\pi]$, let $y_{\theta}(x)$ be a function such that for
all $x>\alpha$ we have $\arg(\Gamma(x+iy_{\theta}(x)))=\theta$. Then
$|\Gamma(x+iy_{\theta}(x))|$ grows at least exponentially to $+\infty$ as $x$
does.

###### Proof.

By (2.7) we know that

$\displaystyle|\Gamma(x+iy_{\theta}(x))|$
$\displaystyle\geq\frac{\sqrt{2\pi}}{e}|z|^{x-\frac{1}{2}}\exp(-y_{\theta}(x)\arg(x+iy_{\theta}(x))-x)$
$\displaystyle=\frac{\sqrt{2\pi}}{e}\exp\left(x(\log|z|-1)-\frac{1}{2}\log|z|-y_{\theta}(x)\arg(x+iy_{\theta}(x))\right).$

By Proposition 2.11 we know that the function $y_{\theta}(x)$ is both
decreasing and positive, which implies that $y_{\theta}(x)$ is bounded as
$x\to+\infty$. On the other hand $\arg(x+iy_{\theta}(x))$ is bounded between
$\left(0,\frac{\pi}{2}\right)$ since $x+iy_{\theta}(x)\in Q_{\alpha,0}$,
therefore $y_{\theta}(x)\arg(x+iy_{\theta}(x))$ is bounded as $x\to+\infty$,
which completes the proof. ∎

###  2.2. Transcendence conjectures of $\Gamma$

For many of the periodic functions we mentioned in the introduction there are
precise conjectures on the geometric conditions needed to ensure that an
algebraic variety $V$ has a Zariski dense intersection with the graph of a
given function $f$. In the case of exponential maps the conditions are usually
referred to as _rotundity_ and _freeness_ (see [BK18, Definition 7.1]),
whereas in the case of automorphic functions the conditions are called
_broadness_ and _freeness_ (or _Hodge-generic_) (see [AK22, Definition 2.3]
for the definitions in the case of the $j$ function, and see [EZ21, §5] for
the general case). The legitimacy of these conditions rests upon well-known
conjectures (such as Schanuel’s conjecture for exponentiation) and important
theorems in functional transcendence (usually referred to as Ax–Schanuel
theorems). No such functional transcendence statement is known for $\Gamma$,
but there are some well-known transcendence conjectures about its values, such
as the following.

######  Conjecture 2.13 (Lang–Rohrlich, see [GMR11, Conjecture 3.3]).

Let $q>2$ be an integer. Then

$\mathrm{tr.deg.}_{\operatorname{\mathbb{Q}}}\operatorname{\mathbb{Q}}\left(\\{\pi\\}\cup\left\\{\Gamma\left(\frac{a}{q}\right):a\in\\{1,\ldots,q\\}\wedge\mathrm{gcd}(a,q)=1\right\\}\right)\geq
1+\frac{\phi(q)}{2},$

where $\phi$ denotes Euler’s totient function.

However, this conjecture does not lend itself to a formulation of geometric
conditions in the same way that Schanuel’s conjecture does for complex
exponentiation. So during the remainder of this section we will present some
suggestions that have been communicated to us by Jonathan Kirby that address
this. The first is a transcendence statement for $\Gamma$ in the spirit of
Schanuel’s conjecture, Conjecture 2.15, and the second is a functional
transcendence conjecture akin to Ax–Schanuel, Conjecture 2.22. They can be
seen as a supplement to the Lang–Rohrlich conjecture as they will avoid
considering rational values for the argument of $\Gamma$, and they serve as a
guideline for the eventual conjecture on the geometric conditions needed for
$\Gamma$ (Conjecture 2.26).

We first remark that Schanuel’s conjecture can be phrased as a comparison
between two notions of dimension: one given by algebraic relations
(transcendence degree), and the other given by
$\operatorname{\mathbb{Q}}$-linear dependencies. Indeed, given a finite subset
$A\subset\mathbb{C}$ we write $\mathrm{l.dim}_{\operatorname{\mathbb{Q}}}(A)$
to denote the linear dimension of the $\operatorname{\mathbb{Q}}$-linear span
generated by the elements of $A$, as a $\operatorname{\mathbb{Q}}$-vector
subspace of $\operatorname{\mathbb{C}}$. Then Schanuel’s conjecture can be
stated as follows.

######  Conjecture 2.14 (Schanuel, see [Lan66, p. 30–31]).

For every $z_{1},\ldots,z_{n}\in\operatorname{\mathbb{C}}$ we have

$\mathrm{tr.deg.}_{\operatorname{\mathbb{Q}}}\operatorname{\mathbb{Q}}(\mathbf{z},\exp(\mathbf{z}))\geq\mathrm{l.dim}_{\operatorname{\mathbb{Q}}}(\mathbf{z}).$

The appearance of the $\operatorname{\mathbb{Q}}$-linear dimension accounts
for the well-known fact that $\operatorname{\mathbb{Q}}$-linear dependencies
among complex numbers are transformed into multiplicative dependencies under
exponentiation.

In the case of $\Gamma$, there are several examples of multiplicative
dependencies among values of $\Gamma$ which result in algebraic numbers. For
example,

(2.9)
$\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)\sqrt[6]{5}\sqrt[4]{5-\frac{7}{\sqrt{5}}+\sqrt{6-\frac{6}{\sqrt{5}}}}=\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)\sqrt{2}\sqrt[20]{3}.$

Identities such as (2.9) are consequences of the three difference equations
above, especially (2.5). The hypotheses in the following conjecture have been
included to prevent the known difference equations from occurring.

######  Conjecture 2.15.

Let $z_{1},\ldots,z_{n}\in\operatorname{\mathbb{C}}$ be such that for every
$i,j\in\\{1,\ldots,n\\}$, with $i\neq j$, we have
$z_{i}\notin\operatorname{\mathbb{Q}}z_{j}+\operatorname{\mathbb{Q}}$. Then

$\mathrm{tr.deg.}_{\operatorname{\mathbb{Q}}}\operatorname{\mathbb{Q}}(\mathbf{z},\Gamma(\mathbf{z}))\geq
n.$

###  2.3. Geometric properties of $\Gamma$

We warn the reader that the terminology defined in this section will be seldom
used later. Instead, the conjectures presented here are intended to provide
context and motivation for the main result.

We recall that the general problem one would like to solve is to find a
minimal set of geometric conditions ensuring that an algebraic variety $V$ has
a Zariski dense intersection with the graph of some transcendental function
$f$. To find the geometric conditions for a given function, in our case
$\Gamma$, one needs to understand the algebraic properties of the function.
For example, even though the complex exponential function is a transcendental
function, it still has algebraic properties, most notably the fact that
$\exp(a+b)=\exp(a)\exp(b)$. This property then has the following consequence:
if $L$ is any $\mathbb{Q}$-linear subspace of $\operatorname{\mathbb{C}}^{n}$
and $\mathbf{a}$ is any point in $\operatorname{\mathbb{C}}^{n}$, then
$L+\mathbf{a}$ is an algebraic subvariety of $\operatorname{\mathbb{C}}^{n}$
with the property that $\exp(L+\mathbf{a})$ is also an algebraic variety.
Borrowing terminology from the world of unlikely intersections, we are looking
for the _weakly special_ varieties (i.e. algebraic varieties which remain
algebraic varieties under the transcendental map). In the case of $\Gamma$,
the difference equations inform what some of those varieties are. We formulate
this more precisely with the following definition.

######  Definition 2.16.

Let $W\subseteq\operatorname{\mathbb{C}}^{n}$ be an algebraic subvariety. We
say that $W$ is $\Gamma$-_weakly special_ if

  1. (a)

$W=\operatorname{\mathbb{C}}^{n}$, or

  2. (b)

$W$ is a proper subvariety of $\operatorname{\mathbb{C}}^{n}$ and the set

(2.10) $\\{(\mathbf{z},\Gamma(\mathbf{z})):\mathbf{z}\in W\\}$

is contained in a proper subvariety of $\operatorname{\mathbb{C}}^{2n}$. In
this case we write $\Gamma_{W}$ to denote the Zariski closure of (2.10).

We say that $W$ is $\Gamma$-_special_ if

  1. (i)

it is weakly special, and

  2. (ii)

if $W$ is a proper subvariety of $\operatorname{\mathbb{C}}^{n}$, then both
$W$ and $\Gamma_{W}$ are defined over $\overline{\mathbb{Q}}$.

In particular $\operatorname{\mathbb{C}}^{n}$ is $\Gamma$-special.

When we work with subvarieties of $\operatorname{\mathbb{C}}^{2n}$ with the
purpose of looking at their possible intersection with the graph of $\Gamma$,
then we will always think of these varieties as defined by polynomials in the
ring $\operatorname{\mathbb{C}}[X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}]$, where
the $X_{i}$ represent the first $n$ coordinates of
$\operatorname{\mathbb{C}}^{2n}$, and the $Y_{i}$ represent the last $n$
coordinates.

We start by considering some trivial examples of $\Gamma$-weakly special
varieties.

######  Example 2.17.

Let $W$ be the diagonal of $\operatorname{\mathbb{C}}^{n}$, that is, $W$ is
defined by

$X_{1}=X_{2}=\cdots=X_{n}.$

Then $W$ is $\Gamma$-special. Indeed, in this case $\Gamma_{W}$ is simply the
subvariety of $\operatorname{\mathbb{C}}^{2n}$ defined by

$X_{1}=X_{2}=\cdots=X_{n}\quad\mbox{ and }\quad Y_{1}=Y_{2}=\cdots=Y_{n}.$

We remark this example is suitable for any (set-theoretic) function.

######  Example 2.18.

Any point $\mathbf{z}\in\operatorname{\mathbb{C}}^{n}$ is trivially
$\Gamma$-weakly special because its image $\Gamma(\mathbf{z})$ is a
0-dimensional algebraic variety. Like in the previous example, this is just a
property of functions.

It would be really interesting to understand the $\Gamma$-special points.
Since $\Gamma(1)=1$, we know that every positive integer $n$ is
$\Gamma$-special because $\Gamma(n)=(n-1)!$. It seems to be unknown if there
are any other $\Gamma$-special points.222For example, it seems to be unknown
whether $\Gamma\left(\frac{1}{5}\right)$ is algebraic.

Now we present an example that is more particular to $\Gamma$.

######  Example 2.19.

The variety $W\subset\operatorname{\mathbb{C}}^{2}$ defined by the equation
$X_{2}=X_{1}+1$ is $\Gamma$-special since $\Gamma_{W}$ is then contained in
the variety defined by

(2.11) $X_{2}=X_{1}+1\quad\mbox{ and }\quad Y_{2}=X_{1}Y_{1}.$

The equation (2.3) shows that $\Gamma_{W}$ is contained in (2.11), and hence
$W$ is at least $\Gamma$-weakly special. But since $\Gamma$ is a
transcendental function, the Zariski closure of the graph
$\\{(z,\Gamma(z)):z\in\operatorname{\mathbb{C}}\setminus\mathbb{Z}^{\leq
0}\\}$ has dimension 2. Therefore, the Zariski closure of $\Gamma_{W}$ must
have dimension at least 2. Since (2.11) has dimension exactly 2 and is defined
over $\mathbb{Q}$, this shows that $W$ is $\Gamma$-special and $\Gamma_{W}$ is
given by (2.11).

The difference equations (2.4) and (2.5) involve the sine and exponential
functions, and as such they do not describe algebraic properties of $\Gamma$,
or in other words, these symmetries of $\Gamma$ cannot be witnessed only using
algebraic varieties the way we did with (2.3) in the last example.

It should be noted that unlike the difference equations satisfied by periodic
functions, such as the exponential maps of semi-abelian varieties or the
uniformization maps of Shimura varieties, the equation (2.3) involves both the
argument and the values of the function. Because of this, our definition of
$\Gamma$-weakly special is about subvarieties of the domain of $\Gamma$,
whereas for the other functions we have mentioned, the definition of weakly
special varieties is about subvarieties of the codomain of the function.

The reason behind our interest in $\Gamma$-weakly special varieties is that
their presence may prevent algebraic varieties from ever intersecting the
graph of $\Gamma$, as the following examples illustrate.

######  Example 2.20.

Define $V\subseteq\operatorname{\mathbb{C}}^{4}$ as

$X_{2}=X_{1}\quad\mbox{ and }\quad Y_{2}=Y_{1}+1.$

then $V$ cannot have points in the graph of any function.

######  Example 2.21.

Define $V\subseteq\operatorname{\mathbb{C}}^{4}$ as

$X_{2}=X_{1}+1\quad\mbox{ and }\quad Y_{2}=X_{1}Y_{1}+1.$

then $V$ cannot have points in the graph of $\Gamma$ as that would contradict
(2.3).

In view of the Lang–Rohrlich Conjecture or Conjecture 2.15, it is not expected
that one can find any other $\Gamma$-weakly special varieties. Both
Lang–Rohrlich and Schanuel’s conjecture remain open, but in the case of $\exp$
we do have a complete understanding of the weakly special varieties. This is
due to Ax’s theorem [Ax71, Theorem 3] (also known as the Ax–Schanuel theorem),
although only a weaker statement is needed (sometimes called the
Ax–Lindemann–Weierstrass theorem). Similarly, we understand the weakly special
varieties of any function for which an Ax–Schanuel theorem is known. We can
then make the following conjecture, which (being functional) could be more
tractable than the other conjectures we have mentioned.

######  Conjecture 2.22 (weak Ax–Schanuel for $\Gamma$).

Let $f_{1},\ldots,f_{n}$ be holomorphic functions such that for every
$i,j\in\\{1,\ldots,n\\}$, with $i\neq j$, we have
$f_{i}\notin\operatorname{\mathbb{Q}}f_{j}+\operatorname{\mathbb{C}}$. Then

$\mathrm{tr.deg.}_{\operatorname{\mathbb{C}}}\operatorname{\mathbb{C}}(\mathbf{f},\Gamma(\mathbf{f}))\geq
n+\mathrm{rank}(\mathrm{Jac}(d\mathbf{f})).$

######  Definition 2.23.

A subvariety $V\subset\operatorname{\mathbb{C}}^{2n}$ is said to be
$\Gamma$-_free_ if it is not contained in any proper subvarieties of the form
$\Gamma_{W}$, where $W$ is $\Gamma$-weakly special.

As we mentioned in Example 2.18, every point is $\Gamma$-weakly special, so
the definition of a $\Gamma$-free variety in particular implies that $V$ has
no constant coordinates.

The weak Ax–Schanuel conjecture for $\Gamma$ tells us that if $V$ has no
constant coordinates and it is not contained in any subvariety of
$\operatorname{\mathbb{C}}^{2n}$ of the form $X_{i}+a_{1}X_{j}+a_{2}=0$, where
$a_{1},a_{2}\in\operatorname{\mathbb{Q}}$, then $V$ is $\Gamma$-free.

Let $n,\ell$ be positive integers with $\ell\leq n$ and
$\mathbf{i}=(i_{1},\ldots,i_{\ell})$ in $\mathbb{N}^{\ell}$ with $1\leq
i_{1}<\ldots<i_{\ell}\leq n$. Define the projection map
$\mathrm{pr}_{\mathbf{i}}:\operatorname{\mathbb{C}}^{n}\rightarrow\operatorname{\mathbb{C}}^{\ell}$
by

$\mathrm{pr}_{\mathbf{i}}(x_{1},\ldots,x_{n}):=(x_{i_{1}},\ldots,x_{i_{\ell}}).$

######  Remark 2.24.

It is easy to see that in the examples we have given above of $W$ being a
$\Gamma$-weakly special subvariety of $\operatorname{\mathbb{C}}^{n}$, for any
choice of indices $1\leq i_{1}<\cdots<i_{\ell}\leq n$ we have that
$\mathrm{pr}_{\mathbf{i}}(W)$ is also a $\Gamma$-weakly special subvariety of
$\operatorname{\mathbb{C}}^{\ell}$.

Define
$\mathrm{Pr}_{\mathbf{i}}:\operatorname{\mathbb{C}}^{2n}\to\operatorname{\mathbb{C}}^{2\ell}$
by

$\mathrm{Pr}_{\mathbf{i}}(\mathbf{x},\mathbf{y}):=(\mathrm{pr}_{\mathbf{i}}(\mathbf{x}),\mathrm{pr}_{\mathbf{i}}(\mathbf{y})).$

The following definition is taken from [AEK21].

######  Definition 2.25.

An algebraic set $V\subseteq\operatorname{\mathbb{C}}^{2n}$ is said to be
_broad_ if for any $\mathbf{i}=(i_{1},\ldots,i_{\ell})$ in $\mathbb{N}^{\ell}$
with $1\leq i_{1}<\ldots<i_{\ell}\leq n$ we have
$\dim\mathrm{Pr}_{\mathbf{i}}(V)\geq\ell$.

In particular, if $V$ is broad then $\dim V\geq n$. We now formulate a
possible answer to the problem of the geometric conditions for $\Gamma$.

######  Conjecture 2.26.

Let $n$ be a positive integer and let
$V\subseteq\operatorname{\mathbb{C}}^{2n}$ be an irreducible subvariety which
is broad and is not contained in any subvariety of the form
$X_{i}+a_{1}X_{j}+a_{2}=0$, where $a_{1},a_{2}\in\operatorname{\mathbb{Q}}$.
Then the intersection of $V$ with the graph of $\Gamma$ is Zariski dense in
$V$.

The conditions in Theorem 1.1 require that $\dim\pi_{1}(V)=n$ and that $V$
have no constant coordinates. This implies that $V$ is broad and that it is
not contained in a $\Gamma$-special variety of the form presented in Example
2.19. Thus, Theorem 1.1 provides evidence for Conjecture 2.26.

##  3\. The case of Plane Curves

In this section we will focus on the case $n=1$ of Theorem 1.1.

###  3.1. A general result about differentially transcendental functions

######  Definition 3.1.

A holomorphic function $f$ is _differentially transcendental_ over
$\operatorname{\mathbb{C}}(z)$ if for every non-negative integer $n$ and every
non-constant polynomial
$p\in\operatorname{\mathbb{C}}[X,Y_{0},Y_{1},\ldots,Y_{n}]$, the function
$p\left(z,f(z),f^{\prime}(z),\ldots,f^{(n)}(z)\right)$ is not identically
zero.

######  Definition 3.2.

An entire function $f(z)$ is said to have _finite order of growth_ if there
are $A,B,\rho\in\mathbb{R}^{+}$ such that for all $z\in\mathbb{C}$ we have

$|f(z)|\leq A\exp\left(B|z|^{\rho}\right).$

######  Theorem 3.3.

Let $f(z)$ be a meromorphic function on $\operatorname{\mathbb{C}}$ with
finitely many poles and let $p(X,Y)\in\operatorname{\mathbb{C}}[X,Y]$ be an
irreducible polynomial which depends at least on $Y$. If $f(z)$ is
differentially transcendental over $\operatorname{\mathbb{C}}(z)$ and has
finite order of growth, then the set
$\\{z\in\operatorname{\mathbb{C}}:p(z,f(z))=0\\}$ is infinite.

###### Proof.

We proceed by contradiction, so suppose that
$\\{z\in\operatorname{\mathbb{C}}:p(z,f(z))=0\\}$ is finite. Then by
Hadamard’s factorization theorem (see e.g. [SS03, Chapter 5, Theorem 5.1]) we
can find a polynomial $q(z)\in\operatorname{\mathbb{C}}[z]$, where the degree
of $q$ is bounded above by the order of growth of $f$, and a rational function
$R(z)\in\operatorname{\mathbb{C}}(z)$ such that

$p(z,f(z))=R(z)\exp(q(z)).$

However, by hypothesis, the left hand side of this equality is differentially
transcendental over $\operatorname{\mathbb{C}}(z)$, whereas the right hand
side is differentially algebraic over $\operatorname{\mathbb{C}}(z)$. ∎

In particular this result immediately applies to the Riemann $\zeta$ function
(and other Dirichlet series), and as such it complements [LS22, Theorem 1.2].
We will use it to give a proof for $\Gamma$ in the next subsection.

###  3.2. The result for $\Gamma$

We will now prove the case $n=1$ of Theorem 1.1. We give two proofs of this
result. The first one uses Theorem 3.3, and on its own does not provide
information about the distribution of the solutions.

The second proof is an explicit method which will allow us to obtain a lower
bound for the number of solutions inside balls of a given radius (see
Corollary 3.8). This proof will illustrate the constructions and methods used
in the proof of Theorem 1.1 in the next section, while avoiding some of the
complications that arise in higher dimensions.

######  Theorem 3.4.

Let $p(X,Y)\in\operatorname{\mathbb{C}}[X,Y]$ be an irreducible polynomial
depending on both variables. Then the set $\\{z\in
Q_{\alpha,0}:p(z,\Gamma(z))=0\\}$ is infinite.

###### First proof of Theorem 3.4.

It is well-known that $\Gamma$ is differentially transcendental (see [Rem98,
Chapter 2, §2.6]). While $\Gamma$ has infinitely many poles (at all non-
positive integers), $\frac{1}{\Gamma}$ is an entire function with order of
growth 1 ([SS03, Chapter 6, Theorem 1.6]). So by Theorem 3.3 we get that the
set $\\{z\in\operatorname{\mathbb{C}}:h\left(z,\frac{1}{\Gamma(z)}\right)\\}$
is infinite for every irreducible polynomial
$h(X,Y)\in\operatorname{\mathbb{C}}[X,Y]$. By clearing denominators for
$\Gamma$, we obtain the result. ∎

Before giving the second proof, we introduce the following definitions.

######  Definition 3.5.

Given a point $\beta\in Q_{\alpha,0}$ and given $R>|\Gamma(\beta)|$, let
$\rho(\beta,R)$ denote the point in $\\{z\in
Q_{\alpha,0}:\arg(\Gamma(z))=\arg(\Gamma(\beta))\\}$ which lies in the same
component as $\beta$, and satisfies $|\Gamma(\rho(\beta,R))|=R$.

We also define $T(\beta,R)$ to be the bounded segment between $\beta$ and
$\rho(\beta,R)$ along the component of $\\{z\in
Q_{\alpha,0}:\arg(\Gamma(z))=\arg(\Gamma(\beta))\\}$ which contains $\beta$.

Given two points $\beta_{1},\beta_{2}\in Q_{\alpha,0}$ satisfying
$|\Gamma(\beta_{1})|=|\Gamma(\beta_{2})|$, define $S(\beta_{1},\beta_{2})$ to
be the bounded segment between $\beta_{1}$ and $\beta_{2}$ within the curve
$C_{|\Gamma(\beta_{1})|}$.

We define $K(\beta,R)$ as the simple closed curve defined by the concatenation
of the following four segments: $S(\beta,\beta^{\ast})$, $T(\beta,R)$,
$T(\beta^{\ast},R)$, $S(\rho(\beta,R),\rho(\beta^{\ast},R))$.

Figure 1. Diagram of the curve $K(\beta,R)$.

###### Second proof of Theorem 3.4.

It suffices to prove that the function $\Gamma(z)-A(z)$ has infinitely many
zeros, where $A(z)$ is a non-constant algebraic function. We may assume that
$A$ is holomorphic on $Q_{\alpha,0}\setminus F$, where $F$ is some compact set
(for the basics of algebraic functions in one variable, see [Ahl21, pp.
284–308]).

Let $D$ be a disc centered at the origin which contains all zeroes and poles
of $A(z)$. To prove the theorem, we will use the Argument Principle.
Specifically, we will construct simple closed curves $K$ in
$Q_{\alpha,0}\setminus D$, so that the winding number of $(\Gamma-A)(K)$
around 0 is nonzero. Since we are avoiding $D$ and $\Gamma$ has no poles in
$Q_{\alpha,0}$, showing that the winding number is nonzero is equivalent in
this case to proving that $\Gamma-A$ has a zero in the region bounded by $K$.

Observe now that for $w\in\operatorname{\mathbb{C}}$ given,
$\lim_{z\to\infty}\left|\frac{A(z+w)}{A(z)}-1\right|=0$, where $z$ tends to
infinity within $Q_{\alpha,0}$, and that the limit is uniform if $w$ is
restricted to a compact set. So there is $N$ such that for all $z\in
Q_{\alpha,0}$ with $|z|\geq N$ and all $w\in\operatorname{\mathbb{C}}$ with
$|w|<3$,

(3.1) $|A(z+w)-A(z)|<\frac{|A(z)|}{4}.$

The set $G:=\\{z\in Q_{\alpha,0}:z\notin D\wedge\Re(z)\geq\max\\{16,N\\}\\}$
is simply connected. The function $z\mapsto|\Gamma(z)|-\frac{|A(z)|}{4}$ is
continuous on $G$. By Corollary 2.4, this function is positive when $\Im(z)$
is fixed and $\Re(z)$ is taken large enough, and by Corollary 2.3 it is
negative when $\Re(z)$ is fixed and $\Im(z)$ is large enough. So there is
$\xi\in G$ such that

(3.2) $|\Gamma(\xi)|=\frac{|A(\xi)|}{4}.$

In particular, $A(\xi)\neq 0$ as $\xi\notin D$. Let $\xi=x+iy$, where
$x,y\in\operatorname{\mathbb{R}}$ and let $r:=|\Gamma(\xi)|$. For each $z\in
S(\xi,\xi^{\ast})$,

$|\Gamma(z)|=r=|\Gamma(\xi)|=\frac{|A(\xi)|}{4}.$

Since $x\geq 16$, Corollary 2.9 shows that for any $z\in S(\xi,\xi^{\ast})$,
we have $|z-\xi|\leq\frac{\pi}{(\log\lfloor 16\rfloor-1)^{2}}<1$. Since also
$x\geq N$, (3.1) tells us that for any $z\in S(\xi,\xi^{\ast})$, we have
$|A(z)-A(\xi))|<r$, i.e., $-A(S(\xi,\xi^{\ast}))\subset B(-A(\xi),r)$.
Similarly, for every $z\in S(\xi,\xi^{\ast})$, $|\Gamma(z)-A(z)-(-A(\xi))|\leq
r+|A(z)-A(\xi)|<2r$, so $(\Gamma-A)(S(\xi,\xi^{\ast}))\subset B(-A(\xi),2r)$.

As we move along $S(\xi,\xi^{\ast})$, the argument of $\Gamma(z)$ cycles once
through $(-\pi,\pi]$, so we may pick $\beta$ in $S(\xi,\xi^{\ast})$ such that

$\arg\Gamma(\beta)=\arg(-A(\xi)).$

For any $z\in S(\beta,\beta^{\ast})$, we have
$|z-\xi|\leq|\beta-\beta^{\ast}|+|\xi-\xi^{\ast}|$, so Corollary 2.9 shows
that $|z-\xi|\leq 2\frac{\pi}{(\log\lfloor 16\rfloor-1)^{2}}<2$. So
$(\Gamma-A)(S(\beta,\beta^{\ast}))\subset B(-A(\xi),2r)$ as well.

Let $\theta:=\arg\Gamma(\beta)=\arg\Gamma(\beta^{\ast})$, and let
$R:=|\Gamma(\xi+1)|$. Now we will consider

$(\Gamma-A)\big{(}T(\beta,R)\cup T(\beta^{\ast},R)\big{)}.$

By (3.1), $-A\big{(}T(\beta,R)\cup T(\beta^{\ast},R)\big{)}\subset
B(-A(\xi),r)$. If $z\in T(\beta,R)\cup T(\beta^{\ast},R)$, then
$(\Gamma-A)(z)\in B(-A(\xi),r)+\Gamma(z)$. We observe that as $z$ moves in
$T(\beta,R)$ or $T(\beta^{\ast},R)$ from $S(\beta,\beta^{\ast})$ to
$S(\rho(\beta,R),\rho(\beta^{\ast},R))$, the set $B(-A(\xi),r)+\Gamma(z)$
moves away from 0. Indeed, along $T(\beta,R)$ and $T(\beta^{\ast},R)$,
$|\Gamma(z)|$ strictly increases and $\arg\Gamma(z)=\arg(-A(\xi))$ is fixed.
So $(\Gamma-A)\big{(}T(\beta,R)\cup T(\beta^{\ast},R)\big{)}$ is confined to
the following region

$U:=\left\\{a+b:a\in B(-A(\xi),r)\wedge\arg b=\theta\wedge r\leq|b|\leq
R\right\\}.$

Next we consider
$(\Gamma-A)\big{(}S(\rho(\beta,R),\rho(\beta^{\ast},R))\big{)}$. The image of
$S\big{(}\rho(\beta,R),\rho(\beta^{\ast},R)\big{)}$ under $\Gamma$ is a circle
of radius $R$ centered at 0 (traversed once). Since

$\displaystyle R$ $\displaystyle=|\xi||\Gamma(\xi)|$
$\displaystyle>12|\Gamma(\xi)|$ by choice of $\Re(\xi)\geq 16$
$\displaystyle=3|A(\xi)|$ by (3.2)
$\displaystyle>2\left(\frac{5}{4}|A(\xi)|\right)$ $\displaystyle>2|A(z)|$
$\displaystyle\text{for all $z\in
S\big{(}\rho(\beta,R),\rho(\beta^{\ast},R)\big{)}$, by \ref{eq:A(z)/4}},$

the image of $S\big{(}\rho(\beta,R),\rho(\beta^{\ast},R)\big{)}$ under
$\Gamma-A$ is contained in the annulus

$V:=\left\\{z\in\operatorname{\mathbb{C}}:\frac{R}{2}<|z|<\frac{3R}{2}\right\\}.$

Since $R>3|A(\xi)|$, $V\cap B(-A(\xi),2r)=\emptyset$. See Figure 2:
$B(-A(\xi),r)$ is represented by the black disk, it is surrounded by the
boundary of $B(-A(\xi),2r)$, the straight lines represent the boundary of $U$,
and $V$ is the annulus determined by the two circles centered at $0$.

Figure 2. Layout of $B(-A(\xi),r)$, $B(-A(\xi),2r)$, $U$ and $V$.

Furthermore, the inequalities above show that for all $z\in
S\big{(}\rho(\beta,R),\rho(\beta^{\ast},R)\big{)}$, we have that
$|\Gamma(z)-(\Gamma-A)(z)|=|A(z)|<\frac{R}{2}$, showing that for all $z\in
S\big{(}\rho(\beta,R),\rho(\beta^{\ast},R)\big{)}$, $(\Gamma-A)(z)$ is in the
ball centered at $\Gamma(z)$ of radius $\frac{R}{2}$.

The image of $K(\beta,R)$ under $\Gamma-A$ must be a closed curve. We know
that

$\displaystyle(\Gamma-A)\big{(}S(\rho(\beta,R),\rho(\beta^{\ast},R))\big{)}\subset$
$\displaystyle V$
$\displaystyle(\Gamma-A)\big{(}S(\beta,\beta^{\ast})\big{)}\subset$
$\displaystyle B(-A(\xi),2r)$ $\displaystyle(\Gamma-A)\big{(}T(\beta,R)\cup
T(\beta^{\ast},R)\big{)}\subset$ $\displaystyle U.$

We also know that $B(-A(\xi),2r)$ and $U$ are connected sets which do not
contain 0. Since for $z\in S\big{(}\rho(\beta,R),\rho(\beta^{\ast},R)\big{)}$,
$\Gamma(z)$ parameterizes the circle of radius $R$ centered at 0 contained in
$V$, and $(\Gamma-A)(z)\subset B\left(\Gamma(z),\frac{R}{2}\right)$, we have
that $(\Gamma-A)\big{(}S(\rho(\beta,R),\rho(\beta^{\ast},R))\big{)}$ does make
at least most of a turn around 0.

So the winding number of $(\Gamma-A)(K(\beta,R))$ around 0 must be equal to
the winding number around 0 of the curve determined by concatenating the
boundaries of $B(-A(\xi),r)$ and $U$ with $\partial B\left(0,R\right)$, which
is 1.

So far we have found one zero of $\Gamma-A$. In order to find more, all we
need to do is choose a new starting point $\xi^{\prime}\in G$ not contained in
the region bounded by $K(\beta,R)$ and repeat the construction above. ∎

######  Remark 3.6.

The diameter of the curve $K(\beta,R)$ used in the previous proof is at most
$\displaystyle\frac{\pi}{(\log\lfloor\Re(\beta)\rfloor-1)^{2}}+1$.

######  Corollary 3.7.

The set $\\{z\in\operatorname{\mathbb{C}}:\Gamma(z)=z\\}$ is Zariski dense in
$\operatorname{\mathbb{C}}$.

###### Proof.

Follows immediately from Theorem 3.4 by considering $p(X,Y):=X-Y$. ∎

######  Corollary 3.8.

Let $A(z)$ be a non-constant algebraic function. For any $\epsilon>0$, there
is an $R_{\epsilon}\in\operatorname{\mathbb{R}}^{+}$ such that for all
$R>R_{\epsilon}$, there are at least $\frac{2R}{1+2\epsilon}-c$ zeroes of
$\Gamma-A$ in the ball $B(0,R)$, where $c$ is a constant depending on $A$.

###### Proof.

We will follow the notation introduced in the proof of Theorem 3.4. Let $N$
and $G$ be as in the proof of Theorem 3.4. Let $b\in G$ be a point of least
modulus such that

$|\Gamma(b)|=\frac{|A(b)|}{4}.$

Let $\displaystyle
c=\frac{2|b|}{1+\frac{2\pi}{(\log\lfloor\Re(b)\rfloor-1)^{2}}}$. Now let
$\epsilon>0$, and let $x=\max\left\\{16,e^{\sqrt{2\pi/\epsilon}+1}\right\\}$.
Then any $z\in G$ with $\Re(z)\geq x$ satisfies
$\displaystyle\frac{2\pi}{(\log\lfloor\Re(z)\rfloor-1)^{2}}<\epsilon$.
Starting from $x$ we will find $\xi\in G$ with $\Re(\xi)\geq x$ satisfying

(3.3) $|\Gamma(\xi)|=\frac{|A(\xi)|}{4}.$

If $|\Gamma(x)|=\frac{|A(x)|}{4}$, then let $\xi=x$. If
$|\Gamma(x)|<\frac{|A(x)|}{4}$, then let $\xi$ be the nearest point on the
horizontal line extending to the right of $x$ satisfying (3.3), which exists
by Corollary 2.4. If $|\Gamma(x)|>\frac{|A(x)|}{4}$, then let $\xi$ be the
nearest point on the vertical line extending up from $x$ satisfying (3.3),
which exists by Corollary 2.3. Moreover, bounds can be computed on how far up
or to the right $\xi$ can be from $x$ using equations (2.8) and (2.7).

Let $K,\beta^{\ast}$ be as in the proof of Theorem 3.4. We will find
$\xi^{\prime}\in G$ with $|\xi^{\prime}|>|\xi|$ and
$|\xi^{\prime}-\xi|<1+2\epsilon$ such that

$|\Gamma(\xi^{\prime})|=\frac{|A(\xi^{\prime})|}{4}$

and the processes in the proof of Theorem 3.4 starting with $\xi$ and
$\xi^{\prime}$ respectively give distinct zeroes of $\Gamma-A$. This shows
that increasing the radius $R$ by $1+2\epsilon$ yields at least one zero of
$\Gamma-A$ in $B(0,R+1+2\epsilon)\setminus B(0,R)$.

First, our choice of $\xi$ gives
$|\beta^{\ast}-\xi|\leq|\beta^{\ast}-\beta|+|\beta-\xi|<2\epsilon$. Note that
for any $w\in G$ with $|w-\beta^{\ast}|<1$, we have

$|A(w)-A(\xi)|<\frac{|A(\xi)|}{4}$

i.e., $\frac{3}{4}|A(\xi)|<|A(w)|<\frac{5}{4}|A(\xi)|$. Since
$|\Gamma(\beta^{\ast})|=\frac{|A(\xi)|}{4}$, we have

$\displaystyle|\Gamma(\beta^{\ast}-1)|$
$\displaystyle=\frac{|\Gamma(\beta^{\ast})|}{|\beta^{\ast}-1|}=\frac{|A(\xi)|}{4|\beta^{\ast}-1|}<\frac{\frac{3}{4}|A(\xi)|}{4}$
$\displaystyle|\Gamma(\beta^{\ast}+1)|$
$\displaystyle=|\beta^{\ast}||\Gamma(\beta^{\ast})|=\frac{|\beta^{\ast}||A(\xi)|}{4}>\frac{\frac{5}{4}|A(\xi)|}{4}.$

Now let $\theta=\arg\Gamma(\beta^{\ast})$, and consider the component
$C\subset\\{z:\arg\Gamma(z)=\theta\\}$ passing through $\beta^{\ast}$. Let
$w_{1}$ be the point on $C$ with $\Re w_{1}<\Re\beta^{\ast}$ such that
$|w_{1}-\beta^{\ast}|=1$, and let $w_{2}$ be the point on $C$ with $\Re
w_{2}>\Re\beta^{\ast}$ such that $|w_{2}-\beta^{\ast}|=1$. Since the rate of
change of $|\Gamma(z)|$ is larger along curves of fixed argument than along
horizontal lines, $|\Gamma(w_{1})|<|\Gamma(\beta^{\ast}-1)|$ and
$|\Gamma(w_{2})|>|\Gamma(\beta^{\ast}+1)|$. So

$\displaystyle|\Gamma(w_{1})|-\frac{|A(w_{1})|}{4}$
$\displaystyle<|\Gamma(w_{1})|-\frac{\frac{3}{4}|A(\xi)|}{4}<|\Gamma(w_{1})|-|\Gamma(\beta^{\ast}-1)|<0$
$\displaystyle|\Gamma(w_{2})|-\frac{|A(w_{2})|}{4}$
$\displaystyle>|\Gamma(w_{2})|-\frac{\frac{5}{4}|A(\xi)|}{4}>|\Gamma(w_{2})|-|\Gamma(\beta^{\ast}+1)|>0$

By the intermediate value theorem, there must be a point $\xi^{\prime}$ in the
segment of $C$ between $w_{1}$ and $w_{2}$ such that

$|\Gamma(\xi^{\prime})|=\frac{|A(\xi^{\prime})|}{4}.$

Let $K^{\prime}$ be the curve and $\beta^{\prime}$ the lower left corner point
constructed via the process in the proof of Theorem 3.4 starting with
$\xi^{\prime}$. Then $T(\beta^{\ast},\rho(\beta^{\ast},R))$, which is the
“top” of $K$, runs along $C$, while
$T(\beta^{\prime},\rho(\beta^{\prime},R^{\prime}))$, which is the “bottom” of
$K^{\prime}$, lies on or above $C$. So the zeroes of $\Gamma-A$ in the regions
bounded by $K$ and $K^{\prime}$ must be distinct.

Finally, since $\overline{\Gamma(z)}=\Gamma(\overline{z})$, reflections of the
argument of Theorem 3.4 and the argument above hold in the lower half quadrant
of $\operatorname{\mathbb{C}}$. So there are actually at least two new zeroes
of $\Gamma-A$ inside the ball $B(0,R+1+2\epsilon)$ compared to $B(0,R)$. ∎

At this point, a natural question to ask in Diophantine geometry is whether
the intersection of an algebraic variety $V$ with the graph of $\Gamma$ can
contain a Zariski dense set of $\Gamma$-special points.333This question is in
the same line as the Manin–Mumford and André–Oort conjectures, which are now
theorems (see [Ray83, McQ95] and [Pil11, PST+21, Tsi18]). As we have
mentioned, we do not have a complete characterization of such points, but at
least we know that all positive integers are $\Gamma$-special. Since
$\Gamma(n)=(n-1)!$, it is a simple matter to show that the intersection of an
algebraic curve $V$ and the graph of $\Gamma$ cannot have infinitely many
integer points due to growth reasons. Furthermore, we have the following
result.

######  Lemma 3.9.

Let $V\subseteq\operatorname{\mathbb{C}}^{2}$ be a free algebraic curve. Then
the set $\\{x\in\operatorname{\mathbb{R}}^{+}:(x,\Gamma(x))\in V\\}$ is
finite.

###### Proof.

If $V$ is not definable over $\operatorname{\mathbb{R}}$, then
$V(\operatorname{\mathbb{R}})$ must be finite, so we only need to consider the
case when $V$ is defined over $\operatorname{\mathbb{R}}$. Since the
restriction of $\Gamma$ to the positive reals is definable in an o-minimal
expansion of $\operatorname{\mathbb{R}}$ [vdDS00], the set
$\\{x\in\operatorname{\mathbb{R}}^{+}:(x,\Gamma(x))\in V\\}$ is either finite
or has positive real dimension. It cannot have positive real dimension because
$\Gamma$ is a transcendental function. ∎

##  4\. Proof of the Main Result

Let $V\subseteq\operatorname{\mathbb{C}}^{2n}$ be an algebraic variety
satisfying the hypotheses of Theorem 1.1. Standard arguments about
intersecting $V$ with generic hyperplanes allow us to reduce to the case where
$\dim V=n$, see e.g. [Asl22, Lemma 4.30] and [Ete22, Proposition 4.4]. To
prove Theorem 1.1 it suffices then to solve systems of equations of the form

(4.1)
$\left\\{\begin{array}[]{ccc}\Gamma(z_{1})&=&A_{1}(z_{1},\ldots,z_{n})\\\
&\vdots&\\\ \Gamma(z_{n})&=&A_{n}(z_{1},\ldots,z_{n})\end{array}\right\\},$

where $A_{1},\ldots,A_{n}$ are algebraic functions determined by $V$. We will
be interested in the behavior of these functions as the $z_{i}$ tend to
$\infty$, and some care is needed to do this correctly.

###  4.1. The general set-up

The following constructions are mostly folklore, but for the sake of
precision, we follow [AKM23, §3].

Consider the complex projective space $\mathbb{P}^{n}$. Given
$\ell\in\\{0,\ldots,n\\}$ define the chart
$U_{\ell}:=\\{[z_{0}:\cdots:z_{n}]\in\mathbb{P}^{n}:z_{\ell}=1\\}$. We now fix
the embedding $\operatorname{\mathbb{C}}^{n}\hookrightarrow\mathbb{P}^{n}$ by
identifying $\operatorname{\mathbb{C}}^{n}$ with $U_{0}$ via the usual map
$(z_{1},\ldots,z_{n})\mapsto[1:z_{1}:\cdots:z_{n}]$.

Given $\ell\in\\{1,\ldots,n\\}$, a point
$\mathbf{c}:=[0:c_{1}:\cdots:c_{n}]\in U_{\ell}\subset\mathbb{P}^{n}$ (where
$c_{\ell}=1$), and $\epsilon>0$, we define the _polydisk of radius $\epsilon$
centered at $\mathbf{c}$_ to be

$D^{\ast}(\mathbf{c},\epsilon):=\\{[z_{0}:\cdots:z_{n}]\in\mathbb{P}^{n}:|z_{0}|<\epsilon\wedge|z_{k}-c_{k}|<\epsilon\mbox{
for }k\in\\{1,\ldots,n\\}\wedge z_{\ell}=1\\}.$

The restriction of this polydisk to $U_{0}$ gives

$D(\mathbf{c},\epsilon):=\left\\{(z_{1},\ldots,z_{n})\in\operatorname{\mathbb{C}}^{n}:|z_{\ell}|>\frac{1}{\epsilon}\wedge\left|\frac{z_{k}}{z_{\ell}}-c_{k}\right|<\epsilon\mbox{
for }k\in\\{1,\ldots,n\\}\right\\}.$

Observe that for $\mathbf{z}\in D(\mathbf{c},\epsilon)$ we have that $z_{k}\in
B(c_{k}z_{\ell},\epsilon|z_{\ell}|)$, for every $k\in\\{1,\ldots,n\\}$. Given
$\theta\in\operatorname{\mathbb{R}}$ and $\eta\in(\theta,\theta+2\pi]$ we
define

$D_{\theta,\eta}(\mathbf{c},\epsilon):=\\{\mathbf{z}\in
D(\mathbf{c},\epsilon):\theta<\arg(z_{\ell})<\eta\\}.$

Since we will be working only on $Q_{\alpha,0}^{n}$, it suffices to take
$\arg(z)$ to always be in $\left(0,\frac{\pi}{2}\right)$. We also define
$D^{\ast}_{\theta,\eta}(\mathbf{c},\epsilon):=D_{\theta,\eta}(\mathbf{c},\epsilon)\cup\left(D^{\ast}(\mathbf{c},\epsilon)\setminus
D(\mathbf{c},\epsilon)\right)$.

Then by [AKM23, Proposition 3.2] we know that for the variety $V$ there is a
Zariski open dense subset $P\subset\mathbb{P}^{n}\setminus U_{0}$ with the
following property: for every $\ell\geq 1$, $\mathbf{c}\in P\cap U_{\ell}$,
$\theta\in\operatorname{\mathbb{R}}$, $\eta\in(\theta,\theta+2\pi]$ and all
sufficiently small polydisks $D^{\ast}(\mathbf{c},\epsilon)$ there are unique
continuous maps
$A_{1},\ldots,A_{n}:D^{\ast}_{\theta,\eta}(\mathbf{c},\epsilon)\to\mathbb{P}^{1}$
such that

  1. (i)

for all $\mathbf{z}\in D_{\theta,\eta}(\mathbf{c},\epsilon)$,
$(\mathbf{z},A_{1}(\mathbf{z}),\ldots,A_{n}(\mathbf{z}))\in V$, and

  2. (ii)

the restriction of each $A_{i}$ to $D_{\theta,\eta}(\mathbf{c},\epsilon)$ is
complex analytic.

######  Remark 4.1.

Choose $\ell\in\\{1,\ldots,n\\}$. Observe that for every $\epsilon>0$ there
are $\epsilon^{\prime},\delta>0$ such that for all
$\mathbf{c},\mathbf{c}^{\prime}\in C\cap U_{\ell}$, if
$\|\mathbf{c}-\mathbf{c}^{\prime}\|<\delta$ then
$D(\mathbf{c}^{\prime},\epsilon^{\prime})\subseteq D(\mathbf{c},\epsilon)$.

Choose $0<\epsilon<\frac{1}{2}$ and choose $\mathbf{c}$ so that $c_{i}\geq 1$
for all $i\in{1,\ldots,n-1}$. Let $A_{1},\ldots,A_{n}$ be the corresponding
algebraic maps defined on $D^{\ast}_{\theta,\eta}(\mathbf{c},\epsilon)$, where
$\theta\in\left(-\pi,-\frac{\pi}{2}\right)$ and
$\eta\in\left(\frac{\pi}{2},\pi\right)$. Since $V$ is free and $P$ is Zariski
dense, we may assume that $A_{i}(c_{1}z,\ldots,c_{n-1}z,z)$ is not identically
zero nor identically $\infty$, for all $i\in\\{1,\ldots,n\\}$.

For the rest of this section, $A_{1},\ldots,A_{n}$, $\mathbf{c}$, $\epsilon$
and $D_{\theta,\eta}(\mathbf{c},\epsilon)$ remain fixed. Given $z_{n}$ with
$|z_{n}|>\frac{1}{\epsilon}$ we have that for every $z_{i}\in
B(c_{i}z_{n},\epsilon|z_{n}|)$, $|z_{i}|<(c_{i}+\epsilon)|z_{n}|$. This shows
that the coordinates of the points in $D(\mathbf{c},\epsilon)$ grow
asymptotically like $|z_{n}|$. So there is $M>\frac{1}{\epsilon}$ such that
for every $i\in\\{1,\ldots,n\\}$ there are
$a_{i},b_{i}\in\operatorname{\mathbb{R}}^{+}$ and $d_{i}\in\mathbb{Z}$ such
that for all $\mathbf{z}\in D_{\theta,\eta}(\mathbf{c},\epsilon)$ with
$|z_{n}|>M$ we have

(4.2) $a_{i}|z_{n}|^{d_{i}-1}<|A_{i}(\mathbf{z})|\leq b_{i}|z_{n}|^{d_{i}}.$

###  4.2. The proofs

Before giving the proof of the main theorem, we present a result that is
stronger in some aspects (as it allows for equations between $\Gamma$ and
other holomorphic functions instead of only algebraic functions) and weaker in
others (as it requires the holomorphic functions to have a non-zero finite
limit at infinity).

######  Proposition 4.2.

Let $f_{1},\ldots,f_{n}:Q_{\alpha,0}^{n}\to\operatorname{\mathbb{C}}$ be
holomorphic functions such that for every $k\in\\{1,\ldots,n\\}$ there is
$w_{k}\in\operatorname{\mathbb{C}}^{\times}$ such that

$\lim_{\mathbf{z}\to\infty}f_{k}(\mathbf{z})=w_{k}.$

Then there are infinitely many solutions to the system of equations

$\displaystyle\Gamma(z_{1})$ $\displaystyle=f_{1}(z_{1},\ldots,z_{n})$
$\displaystyle\vdots$ $\displaystyle\Gamma(z_{n})$
$\displaystyle=f_{n}(z_{1},\ldots,z_{n}).$

###### Proof.

For every $k\in\\{1,\ldots,n\\}$ choose positive real numbers $0<r_{k}<R_{k}$
such that $r_{k}<|w_{k}|<R_{k}$. Let
$\delta:=\min_{k=1,\ldots,n}\\{R_{k}-|w_{k}|,|w_{k}|-r_{k}\\}$, and let $M>0$
be such that if $\mathbf{z}\in Q_{\alpha,0}^{n}$ satisfies $\|\mathbf{z}\|>M$,
then for every $k\in\\{1,\ldots,n\\}$ we have
$|f_{k}(\mathbf{z})-w_{k}|<\delta$. Choose $\boldsymbol{\beta}\in
Q_{\alpha,0}^{n}$ so that $\|\boldsymbol{\beta}\|\geq M$,
$|\Gamma(\beta_{k})|=r_{k}$ and $\arg(\Gamma(\beta_{k}))=\arg(w_{k})-\pi\mod
2\pi$. Let $\Omega_{k}$ be the region bounded by $K(\beta_{k},R_{k})$, and let
$\Omega:=\Omega_{1}\times\cdots\times\Omega_{n}$. We remark that for every
$\mathbf{z}\in\partial\Omega$ there is $k\in\\{1,\ldots,n\\}$ such that
$z_{k}\in K(\beta_{k},R_{k})$.

Now, for every $z\in K(\beta_{k},R_{k})$ one of the following scenarios must
occur:

  1. (a)

$|\Gamma(z)|=r_{k}$, in which case $|\Gamma(z)-w_{k}|\geq|w_{k}|-r_{k}$.

  2. (b)

$\arg(\Gamma(z))=\arg(\Gamma(\beta_{k}))=\arg(w_{k})-\pi$, in which case the
line through $\Gamma(z)$ and $w_{k}$ goes through 0, and 0 is in between
$\Gamma(z)$ and $w_{k}$. Then $|\Gamma(z)-w_{k}|$ is the length of this line
segment, which is at least $|w_{k}|+r_{k}$.

  3. (c)

$|\Gamma(z)|=R_{k}$, in which case $|\Gamma(z)-w_{k}|\geq R_{k}-|w_{k}|$.

So given $\mathbf{z}\in\partial\Omega$ there is $k\in\\{1,\ldots,n\\}$ such
that $|\Gamma(z_{k})-w_{k}|>\delta$, and thus
$|\Gamma(z_{k})-w_{k}|>|f_{k}(\mathbf{z})-w_{k}|$. By the multivariable
version of Rouché’s theorem (see e.g. [Sha92, Theorem 2 in Chapter IV,
§18.55]) we know that the number of zeros of
$(\Gamma(z_{1})-f_{1}(\mathbf{z}),\ldots,\Gamma(z_{n})-f_{n}(\mathbf{z}))$ in
$\Omega$ equals the number of zeros of
$(\Gamma(z_{1})-w_{1},\ldots,\Gamma(z_{n})-w_{n})$ in $\Omega$. By the
construction of $K(\beta_{k},R_{k})$,
$\Gamma(\Omega_{k})=\\{z\in\operatorname{\mathbb{C}}:r_{k}<|z|<R_{k}\\}$, so
there is $z\in\Omega_{k}$ such that $\Gamma(z)=w_{k}$, and thus there is a
zero of $(\Gamma(z_{1})-w_{1},\ldots,\Gamma(z_{n})-w_{n})$ in $\Omega$. We may
now repeat the argument by increasing $\|\boldsymbol{\beta}\|$ to get
infinitely many zeros of
$(\Gamma(z_{1})-f_{1}(\mathbf{z}),\ldots,\Gamma(z_{n})-f_{n}(\mathbf{z}))$. ∎

Now we move towards the proof of Theorem 1.1. We need one lemma first.

######  Lemma 4.3.

Let $A:D_{\theta,\eta}(\mathbf{c},\epsilon)\to\operatorname{\mathbb{C}}$ be an
algebraic function which is continuous on
$D^{\ast}_{\theta,\eta}(\mathbf{c},\epsilon)$. Let
$d\in\operatorname{\mathbb{R}}^{+}$. Then there is
$N\in\operatorname{\mathbb{R}}^{+}$ such that for all $\mathbf{z}\in
D_{\theta,\eta}(\mathbf{c},\epsilon)$ with $\|\mathbf{z}\|>N$ and
$\mathbf{w}\in\operatorname{\mathbb{C}}^{n}$ with $\|\mathbf{w}\|\leq d$
satisfying $\mathbf{z}+\mathbf{w}\in D_{\theta,\eta}(\mathbf{c},\epsilon)$,

$\displaystyle|A(\mathbf{z}+\mathbf{w})-A(\mathbf{z})|<\frac{|A(\mathbf{z})|}{4}.$

###### Proof.

Given $\mathbf{w}\in\operatorname{\mathbb{C}}^{n}$, then by continuity of $A$
we have that

$\lim_{\|\mathbf{z}\|\to+\infty}\left|\frac{A(\mathbf{z}+\mathbf{w})}{A(\mathbf{z})}-1\right|=0,$

where $\mathbf{z}$ only moves within $D_{\theta,\eta}(\mathbf{c},\epsilon)$.
If $\mathbf{w}$ is restricted to move within a compact set, then the limit is
uniform, so there exists $N$ satisfying the conclusion of the lemma. ∎

###### Proof of Theorem 1.1.

Choose $M_{1}>0$ and $j\in\\{1,\ldots,n\\}$ such that for all
$i\in\\{1,\ldots,n\\}$ and all $\mathbf{z}\in
D_{\theta,\eta}(\mathbf{c},\epsilon)$ with $|z_{n}|>M_{1}$,
$|A_{j}(\mathbf{z})|\leq|A_{i}(\mathbf{z})|$. Let $r=\max(16,d_{i}-d_{j}+3)$.
Choose $\boldsymbol{\xi}\in D_{\theta,\eta}(\mathbf{c},\epsilon)$ with
$|\xi_{n}|>M_{1}$ satisfying the following conditions:

  1. (i)

For all $i\in\\{1,\ldots,n\\}$, $\xi_{i}\in Q_{r,r}$ and

$|\xi_{i}|\geq\frac{5b_{i}}{a_{j}(c_{i}-\epsilon)^{d_{i}-d_{j}+1}}.$

  2. (ii)

For all $\mathbf{z}\in\prod_{i=1}^{n}B(\xi_{i},d_{i}-d_{j}+3)$ (here we use
Lemma 4.3)

$|A_{i}(\mathbf{z})-A_{i}(\boldsymbol{\xi})|<\frac{|A_{i}(\boldsymbol{\xi})|}{4}.$

We remark that by the choice of $A_{j}$ we always have that $d_{i}-d_{j}\geq
0$.

  3. (iii)

For all $i\in\\{1,\ldots,n\\}$,
$\displaystyle|\Gamma(\xi_{i})|=\frac{|A_{j}(\boldsymbol{\xi})|}{4}$.

Choose $\beta_{i}\in S(\xi_{i},\xi_{i}^{\ast})$ so that
$\arg(\Gamma(\beta_{i}))=\arg A_{i}(\boldsymbol{\xi})-\pi$. Given
$i\in\\{1,\ldots,n\\}$ consider the curve $K(\beta_{i},R_{i})$, where
$R_{i}=|\Gamma(\beta_{i}+d_{i}-d_{j}+2)|$. Let $\Omega_{i}\subset\mathbb{C}$
be the bounded region defined by $K(\beta_{i},R_{i})$. Define
$\Omega:=\Omega_{1}\times\cdots\times\Omega_{n}$.

For every $\mathbf{z}\in\partial\Omega$ there is $k\in\\{1,\ldots,n\\}$ such
that $z_{k}\in K(\beta_{k},R_{k})$. Observe that in this case we have

  1. (a)

If $z_{k}\in S(\beta_{k},\beta_{k}^{\ast})$, then
$|\Gamma(z_{k})|=|\Gamma(\beta_{k})|=|\Gamma(\xi_{k})|=\frac{|A_{j}(\boldsymbol{\xi})|}{4}\leq\frac{|A_{k}(\boldsymbol{\xi})|}{4}$,
so

$|\Gamma(z_{k})-A_{k}(\boldsymbol{\xi})|\geq\frac{3}{4}|A_{k}(\boldsymbol{\xi})|.$

  2. (b)

If $z_{k}\in T(\beta_{k},R_{k})\cup T(\beta_{k}^{\ast},R_{k})$, then the
points $\Gamma(z_{k})$, $A_{k}(\boldsymbol{\xi})$ and $0$ are collinear, with
0 being in between the other two. Therefore

$|\Gamma(z_{k})-A_{k}(\boldsymbol{\xi})|\geq|A_{k}(\boldsymbol{\xi})|.$

  3. (c)

If $z_{k}\in
S\big{(}\rho(\beta_{k},R_{k}),\rho(\beta_{k}^{\ast},R_{k})\big{)}$, then we
get

$\displaystyle|\Gamma(z_{k})|$ $\displaystyle=R_{k}$
$\displaystyle=|\Gamma(\beta_{k})|\prod_{m=0}^{d_{k}-d_{j}+1}|\beta_{k}+m|$ by
(2.3) $\displaystyle\geq|\Gamma(\xi_{k})||\xi_{k}|^{d_{k}-d_{j}+2}$
$\displaystyle\mbox{since }|\beta_{k}|\geq|\xi_{k}|$
$\displaystyle\geq\frac{|A_{j}(\boldsymbol{\xi})|}{4}\cdot\frac{5b_{k}}{a_{j}(c_{k}-\epsilon)^{d_{k}-d_{j}+1}}\cdot|\xi_{k}|^{d_{k}-d_{j}+1}$
by (i)
$\displaystyle\geq\frac{5b_{k}}{4a_{j}}|A_{j}(\boldsymbol{\xi})||\xi_{n}|^{d_{k}-d_{j}+1}$
$\displaystyle\mbox{since }\boldsymbol{\xi}\in D(\mathbf{c},\epsilon)$
$\displaystyle\geq\frac{5b_{k}}{4a_{j}}a_{j}|\xi_{n}|^{d_{j}-1}|\xi_{n}|^{d_{k}-d_{j}+1}$
by (4.2) $\displaystyle=\frac{5}{4}b_{k}|\xi_{n}|^{d_{k}}$
$\displaystyle\geq\frac{5}{4}|A_{k}(\boldsymbol{\xi})|$ $\displaystyle\mbox{by
(\ref{eq:algfunasympt})}.$

Therefore
$|\Gamma(z_{k})-A_{k}(\boldsymbol{\xi})|\geq\frac{1}{4}|A_{k}(\boldsymbol{\xi})|$.

This shows that for all $\mathbf{z}\in\partial\Omega$ with $z_{k}\in
K(\beta_{k},R_{k})$, we have that

$\displaystyle|\Gamma(z_{k})-A_{k}(\boldsymbol{\xi})|$
$\displaystyle\geq\frac{1}{4}|A_{k}(\boldsymbol{\xi})|$
$\displaystyle>|A_{k}(\mathbf{z})-A_{k}(\boldsymbol{\xi})|$
$\displaystyle\mbox{by (ii)}.$

Observe that the image of $K(\beta_{k},R_{k})$ under $\Gamma$ is the annulus
$\left\\{z\in\operatorname{\mathbb{C}}:\frac{1}{4}|A_{j}(\boldsymbol{\xi})|\leq|z|\leq
R_{k}\right\\}$, so there exists $\boldsymbol{\chi}\in\Omega$ such that
$\Gamma(\boldsymbol{\chi})=(A_{1}(\boldsymbol{\xi}),\ldots,A_{n}(\boldsymbol{\xi}))$.
Therefore by the multivariable version of Rouché’s theorem we have that there
is a point $\mathbf{w}\in\Omega$ such that
$\Gamma(\mathbf{w})=(A_{1}(\mathbf{w}),\ldots,A_{n}(\mathbf{w}))$.

So far we have proven that the intersection of
$V\subset\operatorname{\mathbb{C}}^{2n}$ with the graph of $\Gamma$ is non-
empty for every positive integer $n$ and for every variety $V$ with dominant
projection onto the first $n$ coordinates. Now one can use Rabinowitsch’s
trick to show that this is enough to obtain Zariski density (see [Asl22,
Proposition 4.34] and [Ete22, Lemma 3.8]). Indeed, given a proper subvariety
$Z$ of $V$ one can define an algebraic variety
$V^{\prime}\subseteq\operatorname{\mathbb{C}}^{2n+2}$ with dominant projection
onto the first $n+1$ coordinates and such that the projection onto the first
$n$ coordinates lands in $V\setminus Z$. Using what we have proven, we know
that $V^{\prime}$ contains a point in the graph of $\Gamma$, and therefore so
does $V\setminus Z$. ∎

######  Remark 4.4.

Every point in the intersection between $V$ and the graph of $\Gamma$ is a
solution of a system of the form (4.1). There are only finitely many such
systems for each $V$. For each system we can choose a corresponding domain
$D_{\theta,\eta}(\mathbf{c},\epsilon)$. We can cover
$\left(\mathbb{Z}^{+}\right)^{n}$ by finitely many such domains. By (4.2) we
can conclude that when $V$ satisfies the conditions of Theorem 1.1 and $\dim
V=n$, then the set $\\{(\mathbf{z},\Gamma(\mathbf{z}))\in
V:\mathbf{z}\in\left(\mathbb{Z}^{+}\right)^{n}\\}$ is finite, since when
$\|\mathbf{z}\|$ is large enough, there is some $k\in\\{1,\ldots,n\\}$ for
which $|\Gamma(z_{k})|$ is much bigger than $|A_{k}(\mathbf{z})|$.

Given a function $f$ and a positive integer $n$ we write $f^{\circ n}$ to
denote the composition of $f$ with itself $n$ times. That is $f^{\circ 1}:=f$
and $f^{\circ(m+1)}:=f^{\circ m}\circ f$.

######  Corollary 4.5.

$\Gamma$ has infinitely many periodic points of every period. In other words,
for every positive integer $n$, there are infinitely many points $z$ such that
$\Gamma^{\circ n}(z)=z$ and $\Gamma^{\circ m}(z)\neq z$ for all
$m\in\\{1,\ldots,n-1\\}$.

###### Proof.

Consider the variety

$V:=\left\\{\begin{array}[]{rcc}X_{2}&=&Y_{1}\\\ X_{3}&=&Y_{2}\\\ &\vdots&\\\
X_{1}&=&Y_{n}\end{array}\right\\}.$

By Theorem 1.1 we know that this variety has a Zariski dense intersection with
the graph of $\Gamma$. This proves that there are infinitely many points $z$
satisfying $\Gamma^{\circ n}(z)=z$. If $m\in\\{1,\ldots,n-1\\}$, then the
points which satisfy $\Gamma^{\circ m}(z)=z$ and $\Gamma^{\circ n}(z)=z$ lie
in a proper subvariety of $V$, namely the one cut out by the extra condition
$X_{1}=Y_{m}$. Since there are finitely many possible values of $m$, then by
Zariski density we may find a point with the desired properties. ∎

######  Remark 4.6.

The same argument used to prove Corollary 3.8 shows that given
$\boldsymbol{\xi}$ as in the proof of Theorem 1.1, there are $n$ other nearby
starting points for the argument which lead to distinct points of the form
$(\boldsymbol{w},\Gamma(\boldsymbol{w}))$ in the variety $V$. More precisely,
there are
$\boldsymbol{\xi}_{1}^{\prime},\dots,\boldsymbol{\xi}_{n}^{\prime}\in
D_{\theta,\eta}(\boldsymbol{c},\epsilon)$ with
$\|\boldsymbol{\xi}_{i}^{\prime}\|>\|\boldsymbol{\xi}\|$ and
$\|\boldsymbol{\xi}_{i}^{\prime}-\boldsymbol{\xi}\|<1+\frac{4\pi}{(\log\lfloor\Re(\xi_{i})\rfloor-1)^{2}}$
for $i=1,\dots,n$ which satisfy conditions (i), (ii), and (iii) in the proof
of Theorem 1.1 and produce disjoint regions
$\Omega_{1}^{\prime},\dots,\Omega_{n}^{\prime}\subset
D_{\theta,\eta}(\boldsymbol{c},\epsilon)$, each of which contains a point
$\boldsymbol{w}_{i}$ with
$\Gamma(\boldsymbol{w}_{i})=\big{(}A_{1}(\boldsymbol{w}_{i}),\dots,A_{n}(\boldsymbol{w}_{i})\big{)}$.

## References

  * [AEK21] Vahagn Aslanyan, Sebastian Eterović, and Jonathan Kirby.  Differential existential closedness for the $j$-function.  Proc. Amer. Math. Soc., 149(4):1417–1429, 2021.  https://doi.org/10.1090/proc/15333.
  * [Ahl21] Lars Ahlfors.  Complex analysis—an introduction to the theory of analytic functions of one complex variable.  AMS Chelsea Publishing, Providence, RI, third edition, [2021] ©2021.  Reprint of the 1978 original [0510197].
  * [AK22] Vahagn Aslanyan and Jonathan Kirby.  Blurrings of the $J$-function.  Q. J. Math., 73(2):461–475, 2022.  https://doi.org/10.1093/qmath/haab037.
  * [AKM23] Vahagn Aslanyan, Jonathan Kirby, and Vincenzo Mantova.  A geometric approach to some systems of exponential equations.  Int. Math. Res. Not. IMRN, (5):4046–4081, 2023.
  * [Asl22] Vahagn Aslanyan.  Adequate predimension inequalities in differential fields.  Ann. Pure Appl. Logic, 173(1):Paper No. 103030, 42, 2022.  https://doi.org/10.1016/j.apal.2021.103030.
  * [Ax71] James Ax.  On Schanuel’s conjectures.  Ann. of Math. (2), 93:252–268, 1971.  https://doi.org/10.2307/1970774.
  * [BK18] Martin Bays and Jonathan Kirby.  Pseudo-exponential maps, variants, and quasiminimality.  Algebra Number Theory, 12(3):493–549, 2018.  https://doi.org/10.2140/ant.2018.12.493.
  * [BM17] W. D. Brownawell and D. W. Masser.  Zero estimates with moving targets.  J. Lond. Math. Soc. (2), 95(2):441–454, 2017.  https://doi.org/10.1112/jlms.12014.
  * [DFT18] P. D’Aquino, A. Fornasiero, and G. Terzo.  Generic solutions of equations with iterated exponentials.  Trans. Amer. Math. Soc., 370(2):1393–1407, 2018.
  * [EH21] Sebastian Eterović and Sebastián Herrero.  Solutions of equations involving the modular $j$ function.  Trans. Amer. Math. Soc., 374(6):3971–3998, 2021.
  * [Ete22] Sebastian Eterović.  Generic solution of equations involving the modular $j$-function, 2022\.  Preprint: https://arxiv.org/abs/2209.12192.
  * [EZ21] Sebastian Eterović and Roy Zhao.  Algberaic varieties and automorphic functions.  Preprint: https://arxiv.org/abs/2107.10392, 2021.
  * [Gal21] Francesco Paolo Gallinaro.  Solving systems of equations of raising-to-powers type.  Preprint: https://arxiv.org/abs/2103.15675, 2021.
  * [Gal23a] Francesco Paolo Gallinaro.  Exponential sums equations and tropical geometry.  Selecta Math. (N.S.), 29(4):Paper No. 49, 2023.
  * [Gal23b] Francesco Paolo Gallinaro.  On some systems of equations in abelian varieties.  IMRN, 2023.
  * [GMR11] Sanoli Gun, M. Ram Murty, and Purusottam Rath.  Transcendental nature of special values of $L$-functions.  Canad. J. Math., 63(1):136–152, 2011.
  * [Lan66] Serge Lang.  Introduction to transcendental numbers.  Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966\.
  * [LS22] Bao Qin Li and Jörn Steuding.  Fixed points of the Riemann zeta function and Dirichlet series.  Monatsh. Math., 198(3):581–589, 2022.
  * [Mar06] David Marker.  A remark on Zilber’s pseudoexponentiation.  J. Symbolic Logic, 71(3):791–798, 2006.  https://doi.org/10.2178/jsl/1154698577.
  * [McQ95] Michael McQuillan.  Division points on semi-abelian varieties.  Invent. Math., 120(1):143–159, 1995.
  * [MM23] Vincenzo Mantova and David Masser.  Polynomial-exponential equations – some new cases of solvability, 2023\.  Preprint: https://arxiv.org/abs/2303.05592.
  * [Pil11] Jonathan Pila.  O-minimality and the André-Oort conjecture for $\mathbb{C}^{n}$.  Ann. of Math. (2), 173(3):1779–1840, 2011.  https://doi.org/10.4007/annals.2011.173.3.11.
  * [PST+21] Jonathan Pila, Ananth N. Shankar, Jacob Tsimerman, Héène Esnault, and Michael Groechenig.  Canonical heights on shimura varieties and the André-Oort conjecture, 2021.  Preprint: https://arxiv.org/abs/2109.08788.
  * [Ray83] M. Raynaud.  Courbes sur une variété abélienne et points de torsion.  Invent. Math., 71(1):207–233, 1983.
  * [Rem98] Reinhold Remmert.  Classical topics in complex function theory, volume 172 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1998.  Translated from the German by Leslie Kay.
  * [Sha92] B. V. Shabat.  Introduction to complex analysis. Part II, volume 110 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, russian edition, 1992.  Functions of several variables.
  * [SS03] Elias M. Stein and Rami Shakarchi.  Complex analysis, volume 2 of Princeton Lectures in Analysis.  Princeton University Press, Princeton, NJ, 2003.
  * [Tsi18] Jacob Tsimerman.  The André-Oort conjecture for $\mathcal{A}_{g}$.  Ann. of Math. (2), 187(2):379–390, 2018.
  * [Uch12] Mitsuru Uchiyama.  The principal inverse of the gamma function.  Proc. Amer. Math. Soc., 140(4):1343–1348, 2012.
  * [vdDS00] Lou van den Dries and Patrick Speissegger.  The field of reals with multisummable series and the exponential function.  Proc. London Math. Soc. (3), 81(3):513–565, 2000.
  * [Zil02] Boris Zilber.  Exponential sums equations and the Schanuel conjecture.  J. London Math. Soc. (2), 65(1):27–44, 2002.
  * [Zil05] Boris Zilber.  Pseudo-exponentiation on algebraically closed fields of characteristic zero.  Ann. Pure Appl. Logic, 132(1):67–95, 2005.  https://doi.org/10.1016/j.apal.2004.07.001.