# A truncated second main theorem for algebraic tori
with moving targets and applications

Ji Guo  Institute of Mathematics
Academia Sinica
6F, Astronomy-Mathematics Building
No. 1, Sec. 4, Roosevelt Road
Taipei 10617
Taiwan<EMAIL_ADDRESS>,  Chia-Liang Sun  Institute of Mathematics
Academia Sinica
6F, Astronomy-Mathematics Building
No. 1, Sec. 4, Roosevelt Road
Taipei 10617
Taiwan<EMAIL_ADDRESS>and  Julie Tzu-Yueh Wang  Institute of
Mathematics
Academia Sinica
6F, Astronomy-Mathematics Building
No. 1, Sec. 4, Roosevelt Road
Taipei 10617
Taiwan<EMAIL_ADDRESS>
###### Abstract.

We establish a second main theorem for algebraic tori with slow growth moving
targets with truncation to level 1. As the first application of this result,
we prove the Green-Griffith-Lang conjecture for projective spaces with $n+1$
components in the context of moving targets of slow growth. Then we discuss
the integrability of the ring of exponential polynomials in the ring of entire
functions as another application.

2010 Mathematics Subject Classification. Primary 30D35, Secondary 30A70,11J97.

The second named was supported by Taiwan’s MoST grant 109-2811-M-001-613 and
the third named author was supported in part by Taiwan’s MoST grant
108-2115-M-001-001-MY2.

##  1\. Introduction

Originating in the work of Osgood, Vojta, and Lang, it has been observed that
there is a striking correspondence between many statements in Nevanlinna
theory and statements in Diophantine approximation. A detailed “dictionary”
between the two subjects has been constructed by Vojta [18]. The following
conjecture can be viewed as the complex analogue of Vojta’s generalized abc
conjecture ([20, Conjecture 23.4]).

######  Conjecture 1.1.

Let $X$ be a smooth complex projective variety, let D be a normal crossing
divisor on $X$, let $K$ be the canonical divisor on $X$, and let $A$ be an
ample divisor on $X$. Then:

  1. (a)

If $f:\mathbb{C}\to X$ be an algebraically nondegenerate analytic map, then

(1) $\displaystyle N_{f}^{(1)}(D,r)\geq_{\operatorname{exc}}T_{K+D,f}(r)-{\rm
o}(T_{A,f}(r)).$

  2. (b)

For any $\epsilon>0$, there is a proper Zariski-closed subset $Z$ of $X$,
depending only on $X$, $D$, $A$, and $\epsilon$ such that for any analytic map
$f:\mathbb{C}\to X$ whose image is not contain in $Z$, the following

(2) $\displaystyle
N_{f}^{(1)}(D,r)\geq_{\operatorname{exc}}T_{K+D,f}(r)-\epsilon T_{A,f}(r)$

holds.

Here, for positive integer $n$, $N_{f}^{(n)}(D,r)$ is the $n$-truncated
counting function with respect to $D$ is given by

(3) $\displaystyle N_{f}^{(n)}(D,r)=\sum_{0<|z|<r}\min\\{{\rm
ord}_{z}f^{*}D,n\\}\log\frac{r}{|z|}+\min\\{{\rm ord}_{0}f^{*}D,n\\}\log r,$

$T_{D,f}(r)$ is the (Nenvanlinna) height function relative to the divisor $D$
(See [20, Section 12].), and the notion $\leq_{\operatorname{exc}}$ means that
the estimate holds for all $r$ outside a set of finite Lebesgue measure.
Unlike the situation in number fields, there are some known results for this
conjecture. For example, the conjecture holds for $\dim X=1$. The Cartan’s
second main theorem, where $X=\mathbb{P}^{n}$ and $D=H_{1}+\cdots+H_{q}$,
where the $H_{i}$ are hyperplanes in general position, suggests that (1) holds
with $N_{f}^{(1)}(D,r)$ replaced by $\sum_{i=1}^{q}N_{f}^{(n)}(H_{i},r)$ under
a weaker assumption that the map $f$ is linearly non-degenerate. When $X$ is a
semiabelian variety, Noguchi, Winkleman and Yamanoi in [14] showed that the
inequality (1) holds with $N_{f}^{(1)}(D,r)$ replaced by
$N_{f}^{(k_{0})}(D,r)$ for some positive integer $k_{0}$, and (2) holds if the
map is algebraically nondegenerate.

The above conjecture is much harder for the case of moving targets, i.e. the
divisor $D$ is defined over a field of “small functions” with respect to the
map $f$. The only existing result in the moving case is due to Yamanoi in [22]
for $\dim X=1$. All the other results in this direction are stated with a very
high truncated level.(See Theorem 2.6, which is a result of Dethloff and Tan,
for an example.) Our first result is to establish the inequality (2) for
complex tori with moving targets. We recall that the small field with respect
to a holomorphic curve $\mathbf{f}:\mathbb{C}\to\mathbb{P}^{n}(\mathbb{C})$ is
given by

(4) $K_{\mathbf{f}}:=\\{a:a\text{ is a meromorphic function with
}T_{a}(r)={\rm o}(T_{\mathbf{f}}(r))\\}.$

######  Theorem 1.2.

Let $u_{0},u_{1},\dots,u_{n}$ be nonconstant entire functions without zeros,
i.e. $\mathbf{u}=(u_{0},\ldots,u_{n}):\mathbb{C}\to\mathbb{G}_{m}^{n+1}$. Let
$K_{\mathbf{u}}$ be the small field with respect to $\mathbf{u}$. Let $G$ be a
nonconstant homogeneous polynomials in $K_{\mathbf{u}}[x_{0},\ldots,x_{n}]$
with no repeated nonmonomial factors in $K_{\mathbf{u}}[x_{0},\ldots,x_{n}]$.
If $u_{0},\dots,u_{n}$ are multiplicatively independent modulo
$K_{\mathbf{u}}$, then for any $\epsilon>0$

(5) $\displaystyle
N_{G(\mathbf{u})}(0,r)-N^{(1)}_{G(\mathbf{u})}(0,r)\leq_{\operatorname{exc}}\epsilon
T_{\mathbf{u}}(r).$

If we assume furthermore that none of the functions $G(1,0,\dots,0),\dots,$
$G(0,\dots,0,1)$ is identically zero, then

(6) $\displaystyle N^{(1)}_{G(\mathbf{u})}(0,r)\geq_{\rm exc}(\deg
G-\epsilon)\cdot T_{\mathbf{u}}(r).$

Moreover, if each $u_{j}$, $0\leq j\leq n$, is of finite order and
$G_{i}\in\mathbb{C}[z][x_{0},\ldots,x_{n}]$, $1\leq i\leq q$, then the both
assertions above hold under the weaker assumption that $u_{1},\ldots,u_{n}$
are multiplicatively independent modulo $\mathbb{C}$.

Here, $T_{\mathbf{u}}(r)$ is the Nevanlinna height function associated to
${\mathbf{u}}$ and $N_{G(\mathbf{u})}(0,r)$ ($N^{(1)}_{G(\mathbf{u})}(0,r)$
respectively) is the counting function associated to 0 and $G(\mathbf{u})$
(with truncation to level $1$ respectively) to be defined in the next session.
The first application of Theorem 1.2 concerns a case of the Green-Griffith-
Lang conjecture for projective spaces in the context of moving targets of slow
growth. The Green-Griffiths-Lang conjecture in the non-compact case (see [20,
Proposition 15.3]) reads as follows: If $X$ is a smooth variety, $D$ is a
normal crossing divisor on $X$, and $X\setminus D$ is a variety of log general
type, then a holomorphic map $f:\mathbb{C}\to X\setminus D$ cannot have
Zariski-dense image. When $X=\mathbb{P}^{n}$, the condition for $X\setminus D$
to be of log general type is equivalent to the inequality $\deg D\geq n+2$. In
this setting, the conjecture is verified by Green [6] when $D$ has at least
$n+2$ components, and when $n=2$ of 3 components with $\deg D=4$ in [5] under
the assumption that $f$ is of finite order. The $n+1$ component case (with
$\deg D\geq n+2$) is solved by Noguchi, Winkelman and Yamanoi in [13, Theorem
5.4]. For the moving target situation, the $n+2$ component case follows
directly from the second main theorem for moving hypersurfaces in [4]; the
boundary case of $n+1$ components, i.e. $\deg D=n+2$, is recently established
in [9]. The following theorem gives a full treatment to the case of $n+1$
components.

######  Theorem 1.3.

Let $\mathbf{f}=(f_{0},\ldots,f_{n}):\mathbb{C}\to\mathbb{P}^{n}$ be a
holomorphic map, where $f_{0},\ldots,f_{n}$ are entire functions without
common zeros. Let $F_{i}$, $1\leq i\leq n+1$ be homogeneous irreducible
polynomials of positive degree in $K_{\mathbf{f}}[x_{0},\ldots,x_{n}]$ such
that $\sum_{i=1}^{n+1}\deg F_{i}\geq n+2$. Assume that there exists
$z_{0}\in\mathbb{C}$ such that all the coefficients of all $F_{i}$, $1\leq
i\leq n+1$ are holomorphic at $z_{0}$ and the zero locus of $F_{i}$, $1\leq
i\leq n+1$, evaluating at $z_{0}$ intersect transversally. If
$F_{i}(\mathbf{f})$, $1\leq i\leq n+1$, are entire function without zeros,
then $\mathbf{f}$ is algebraically degenerate over $K_{\mathbf{f}}$. Moreover,
if $\mathbf{f}$ is of finite order and
$F_{i}\in\mathbb{C}[x_{0},\ldots,x_{n}]$ for each $i$, then $\mathbf{f}$ is
algebraically degenerate over $\mathbb{C}$.

The second application of Theorem 1.2 is related to an algebraic problem of
exponential polynomials. We recall the definition of exponential polynomials
of order $q\in\mathbb{N}$ as follows.

######  Definition 1.4.

An exponential polynomial of order $q$ is an entire function of the form

(7) $\displaystyle f(z)=P_{1}(z)e^{Q_{1}(z)}+\cdots+P_{m}(z)e^{Q_{m}(z)},$

where $m\in\mathbb{N}$, $P_{i},Q_{i}\in\mathbb{C}[z]$, $1\leq i\leq m$, such
that

$\max_{1\leq i\leq m}\\{\deg Q_{i}\\}=q.$

Our interest of studying exponential polynomials comes from its correspondence
with linear recurrences and its algebraic structure. In [9], we showed that if
an exponential polynomial $f$ is a $d$-th power of some entire function $g$,
i.e. $f=g^{d}$, then $g$ is also an exponential polynomial. In view of the
correspondence between Diophatine approximation and Nevanlinna theory, this
result is a complex analogue of Pisot’s $d$-th root conjecture for linear
recurrences, which is completely solved in [23]. From the view points of
studying the algebraic structure of the exponential polynomials, it basically
says that the radical of an exponential polynomial is also an exponential
polynomial. We refer to [10], [7], and [8] for other expositions in this
direction; and [2] for related results in logic. The following is a nature
question on the integrality of the ring of exponential polynomials over the
ring of entire functions.

######  Problem.

Let $q$ and $d$ be positive integers. Denote by $\mathcal{K}_{q}$ the quotient
field of the ring $\mathcal{E}_{q}$ of the exponential polynomials of order at
most $q$. Let $A_{0},\ldots,A_{d-1}\in\mathcal{E}_{q}$ such that the
polynomial $F(Y):=Y^{d}+A_{d-1}Y^{d-1}+\ldots+A_{1}Y+A_{0}$ is irreducible
over $\mathcal{K}_{q}[Y]$. Suppose that there exists an entire function $g$
such that $F(g)=0$. Is it true that $g\in\mathcal{E}_{q}$?

For the case $q=1$ and each $A_{i}$ is an exponential polynomial with constant
coefficients, refer to [15] for a complete solution.

Being not able to solve the problem completely, we prove the following generic
result.

######  Theorem 1.5.

Let $d$ and $q_{i}$, $1\leq i\leq n$ be positive integers and let
$q=\max\\{q_{1},\ldots,q_{n}\\}$. Let
$\mu_{1},\ldots,\mu_{n}\in\mathbb{C}^{*}$ and
$\mathbf{u}=(u_{1},\ldots,u_{n})=(e^{\mu_{1}z^{q_{1}}},\ldots,e^{\mu_{n}z^{q_{n}}})$.
Assume that $u_{1},\ldots,u_{n}$ are multiplicatively independent modulo
$\mathbb{C}$. Let $A_{i}\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$,
$0\leq i\leq d-1$. Assume that the polynomial
$F(Y):=Y^{d}+A_{d-1}Y^{d-1}+\ldots+A_{1}Y+A_{0}\in\mathcal{K}_{q}[Y]$ is
irreducible and its discriminant $\Delta$ is square-free in
$\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$, i.e. $\Delta$ has no non-
unit repeated factor. Suppose that there exists an entire function $g$ such
that $F(g)=0$. Then $g\in\mathcal{E}_{q}$.

Let us explain how this theorem is related to the problem above. Let
$A_{0},\ldots,A_{d-1}\in\mathcal{E}_{q}$ such that at least one of the $A_{i}$
is of order $q$. Then we can find units of finite order $u_{1},\ldots,u_{n}$,
which are multiplicative independent modulo $\mathbb{C}$, equivalently
algebraic independent over $\mathbb{C}(z)$, as in Theorem 1.5 such that each
$A_{i}\in\mathbb{C}[z][u_{1}^{\pm 1},\ldots,u_{n}^{\pm
1}]\subset\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$. Note that the
latter ring is a unique factorization domain. We refer to the proof of Theorem
1.3 in [9] for such constructions. Theorem 1.5 then applies to the problem
under this identification.

Our proof of Theorem 1.2 relies on the GCD theorem [12] of Levin and the third
author and the machinery developed in [9]. The proof of Theorem 1.3 follows
the ideas in [13] and [3] with extension to the moving situation. We note that
Capuano and Turchet in [1] generalized the work of [3] to non-split function
fields (i.e. the moving case in our terminology) for surfaces. Finally, the
proof of Theorem 1.5 is an adaption and generalization of Theorem 3 in [3] to
the complex situation.

Some background materials will be given in the next session. The key lemmas
are stated in Session 3. The proofs of our theorems will be given in Section
4-6 respectively.

##  2\. Preliminaries

We will give relevant materials and derive some basic results in this session.

###  2.1. Nevanlinna Theory

We will set up some notation and definitions in Nevanlinna theory and recall
some basic results. We refer to [11] and [16] for details.

Let $f$ be a meromorphic function and $z\in\mathbb{C}$ be a complex number.
Denote $v_{z}(f):=\operatorname{ord}_{z}(f)$,

$v_{z}^{+}(f):=\max\\{0,v_{z}(f)\\},\quad\text{and }\quad
v_{z}^{-}(f):=-\min\\{0,v_{z}(f)\\}.$

Let $n_{f}(\infty,r)$ (respectively, ${n}^{(Q)}_{f}(\infty,r)$) denote the
number of poles of $f$ in $\\{z:|z|\leq r\\}$, counting multiplicity
(respectively, ignoring multiplicity larger than $Q\in\mathbb{N}$). The
counting function and truncated counting function of $f$ of order $Q$ at
$\infty$ are defined respectively by

$\displaystyle N_{f}(\infty,r)$
$\displaystyle:=\int_{0}^{r}\frac{n_{f}(\infty,t)-n_{f}(\infty,0)}{t}dt+n_{f}(\infty,0)\log
r$ $\displaystyle=\sum_{0<|z|\leq
r}v_{z}^{-}(f)\log|\frac{r}{z}|+v_{0}^{-}(f)\log r,$

and

$\displaystyle N^{(Q)}_{f}(\infty,r)$
$\displaystyle:=\int_{0}^{r}\frac{n^{(Q)}_{f}(\infty,t)-n^{(Q)}_{f}(\infty,0)}{t}dt+n^{(Q)}_{f}(\infty,0)\log
r$ $\displaystyle=\sum_{0<|z|\leq
r}\min\\{Q,v_{z}^{-}(f)\\}\log|\frac{r}{z}|+\min\\{Q,v_{0}^{-}(f)\\}\log r.$

Then define the counting function $N_{f}(r,a)$ and the truncated counting
function $N^{(Q)}_{f}(r,a)$ for $a\in\mathbb{C}$ as

$N_{f}(a,r):=N_{1/(f-a)}(r,\infty)\quad\text{and}\quad
N^{(Q)}_{f}(a,r):=N^{(Q)}_{1/(f-a)}(\infty,r).$

The proximity function $m_{f}(\infty,r)$ is defined by

$m_{f}(\infty,r):=\int_{0}^{2\pi}\log^{+}|f(re^{i\theta})|\frac{d\theta}{2\pi},$

where $\log^{+}x=\max\\{0,\log x\\}$ for $x\geq 0$. For any $a\in\mathbb{C},$
the proximity function $m_{f}(a,r)$ is defined by

$m_{f}(a,r):=m_{1/(f-a)}(\infty,r).$

The characteristic function is defined by

$T_{f}(r):=m_{f}(\infty,r)+N_{f}(\infty,r).$

It satisfies the First Main Theorem as follows.

######  Theorem 2.1.

Let $f$ be a non-constant meromorphic function on $\mathbb{C}$. Then for every
$a\in\mathbb{C}$ and for any positive real number $r$,

$m_{f}(a,r)+N_{f}(a,r)=T_{f}(r)+O(1).$

where $O(1)$ is independent of $r$.

Rational functions over $\mathbb{C}$ can be characterized by characteristic
functions as follows. (See[11, Chapter VI, Theorem 2.6].)

######  Theorem 2.2.

Let $f$ be a meromorphic function on $\mathbb{C}$. Then $f$ is a rational
function, i.e. $f\in\mathbb{C}(z)$, if and only if $T_{f}(r)=O(\log r)$ for
$r\to\infty$.

We recall the lemma on the logarithmic derivative. (See [16, Theorem A1.2.5].)

######  Lemma 2.3.

Let $f$ be a non-constant meromorphic function on $\mathbb{C}$. Then for any
$\varepsilon>0$,

$m_{f^{\prime}/f}(\infty,r)\leq_{\operatorname{exc}}\log
T_{f}(r)+(1+\varepsilon)\log^{+}\log T_{f}(r)+O(1).$

Let ${\mathbf{f}}:\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a
holomorphic map and $(f_{0},\dots,f_{n})$ be a reduced representation of
${\mathbf{f}}$, i.e. $f_{0},\dots,f_{n}$ are entire functions on $\mathbb{C}$
without common zeros. The Nevanlinna-Cartan characteristic function
$T_{\mathbf{f}}(r)$ is defined by

$T_{\mathbf{f}}(r)=\int_{0}^{2\pi}\log\max\\{|f_{0}(re^{i\theta})|,\dots,|f_{n}(re^{i\theta})|\\}\frac{d\theta}{2\pi}+O(1).$

This definition is independent, up to an additive constant, of the choice of
the reduced representation of ${\mathbf{f}}$.

We will make use of the following elementary inequality . (See [12, Lemma 2.5]
for a proof.)

######  Lemma 2.4.

Let $g_{1},\ldots,g_{n}$ be meromorphic functions. Let ${\bf
g}:=(1,g_{1},\dots,g_{n}):\mathbb{C}\to\mathbb{P}^{n}.$ Then

$\displaystyle T_{g_{i}}(r)\leq T_{\bf
g}(r)\leq\sum_{i=1}^{n}T_{g_{i}}(r)+O(1),$

for $1\leq i\leq n$.

We will use the following version of a truncated second main theorem.

######  Theorem 2.5 ([17, Theorem 2.1]).

Let $\mathbf{f}=(f_{0},\ldots,f_{n}):\mathbb{C}\to\mathbb{P}^{n}(\mathbb{C})$
be a holomorphic map with $f_{0},\dots,f_{n}$ entire and no common zeros.
Assume that $f_{n+1}$ is a holomorphic function satisfying the equation
$f_{0}+\dots+f_{n}+f_{n+1}=0$. If $\sum_{i\in I}f_{i}\neq 0$ for any proper
subset $I\subset\\{0,\dots,n+1\\}$, then

$T_{\mathbf{f}}(r)\leq_{\operatorname{exc}}\sum_{i=0}^{n+1}N_{f_{i}}^{(n)}(0,r)+O(\log^{+}T_{\mathbf{f}}(r)).$

We now recall the following second main theorem for hypersurfaces with moving
targets from [4]. Let
${\mathbf{f}}:\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a
holomorphic map. We denote by

$K_{\mathbf{f}}:=\\{a:a\text{ is a meromorphic function with
}T_{a}(r)=o(T_{\mathbf{f}}(r))\\}.$

A set $\\{Q_{1},\dots,Q_{q}\\}$ of homogeneous polynomials in
$K_{\mathbf{f}}[x_{0},\dots,x_{n}]$ is said to be in weakly general position
if there exists $z_{0}\in\mathbb{C}$ in which all coefficient functions of all
$Q_{j}$, $j=1,\dots,q$ are holomorphic and such that for any $1\leq
j_{0}<\cdots<j_{n}\leq q$ the system of equations
$\left\\{Q_{j_{t}}(z_{0})(x_{0},\dots,x_{n})=0:0\leq t\leq n\right\\}$ has
only the trivial solution
$(x_{0},\dots,x_{n})=(0,\dots,0)\in\mathbb{C}^{n+1}$.

The following statement is a direct consequence of the main theorem in [4].

######  Theorem 2.6 ([4]).

Let $\mathbf{f}$ be a nonconstant meromorphic map of $\mathbb{C}$ into
$\mathbb{P}^{n}$. Let $K\subset K_{\mathbf{f}}$ be a subfield. Let
$\\{Q_{j}\\}_{j=1}^{q}$ be a set of homogeneous polynomials in
$K[x_{0},\dots,x_{n}]$ in weakly general position and with $\deg
Q_{j}=d_{j}\geq 1$. Assume that $\mathbf{f}$ is algebraically non-degenerate
over $K$. Then for any $\epsilon>0$, there exists a positive integer $L$
depending only on $n$, $\epsilon$ and $d_{j}$, $1\leq j\leq q$, such that the
following inequality holds:

$(q-n-1-\epsilon)T_{\mathbf{f}}(r)\leq_{\operatorname{exc}}\sum_{j=1}^{q}\frac{1}{d_{j}}N_{Q_{j}(\mathbf{f})}^{(L)}(0,r).$

We note that the integer $L$ in this theorem is large as mentioned in [4,
Proposition 1.2].

We will need the following version of Hilbert Nullstellensatz reformulated
from [4, Proposition 2.1]. (See also [21, Chapter XI].)

######  Proposition 2.7 ([4, Proposition 2.1]).

Let $\mathbf{f}$ be a nonconstant meromorphic map of $\mathbb{C}$ into
$\mathbb{P}^{n}$. Let $K\subset K_{\mathbf{f}}$ be a subfield. Let
$\\{Q_{i}\\}_{i=1}^{n+1}$ be a set of homogeneous polynomials in
$K[x_{0},\dots,x_{n}]$ in weakly general position and with $\deg
Q_{j}=d_{j}\geq 1$. Then there exists a positive integer $s$, $R\in K$ not
identically zero and $P_{ji}\in K[x_{0},\dots,x_{n}]$, $1\leq i,j\leq n+1$,
such that

$x_{j}^{s}\cdot R=\sum_{i=1}^{n+1}P_{ji}Q_{i}$

for each $0\leq j\leq n.$

Finally, we recall the following definitions. A meromorphic function $f$ is of
finite order (or more exactly, of order $q$) if

$\limsup_{r\to\infty}\frac{\log T_{f}(r)}{\log r}=q.$

In addition, if $f$ is a unit, i.e. an entire function without zeros, of order
$q$, then $u=e^{P}$ for some $P\in\mathbb{C}[z]$ of degree $q$. Similarly, a
map $\mathbf{f}:\mathbb{C}\to\mathbb{P}^{n}(\mathbb{C})$ is said to be of
finite order (or more exactly, of order $q$) if

$\limsup_{r\to\infty}\frac{\log T_{\mathbf{f}}(r)}{\log r}=q.$

###  2.2. GCD in Nevanlinna theory

We recall the gcd counting function of two meromorphic functions and a gcd
theorem with moving targets from [12].

Let $f$ and $g$ be meromorphic functions. We let

$n(f,g,r):=\sum_{|z|\leq r}\min\\{v_{z}^{+}(f),v_{z}^{+}(g)\\}$

and

$N_{\gcd}(f,g,r):=\int_{0}^{r}\frac{n(f,g,t)-n(f,g,0)}{t}dt+n(f,g,0)\log r.$

The gcd theorem we need below is a little more general than what is stated in
[12, Theorem 5.1], and it follows clearly from their proof.

######  Theorem 2.8 ([12, Theorem 5.1]).

Let $u_{1},\dots,u_{n}$ be entire functions without zeros. Let $K\subset
K_{\mathbf{u}}$ be a subfield, and let $F,G\in K[x_{1},\dots,x_{n}]$ be
nonconstant coprime polynomials. Then for every $\varepsilon>0$ the following
inequality holds

$N_{\gcd}(F(u_{1},\dots,u_{n}),G(u_{1},\dots,u_{n}),r)\leq_{\operatorname{exc}}\varepsilon\max_{1\leq
j\leq n}\\{T_{u_{j}}(r)\\},$

if $u_{1},\dots,u_{n}$ are algebraically independent over $K$.

######  Remark.

The proof of [12, Theorem 5.1] actually only treat the case where $n\geq 2$
because the case where $n=1$ is much easier, as shown in the following
argument below. Let $u$ be an entire function without zero. Suppose that $F$
and $G$ are coprime polynomials in $K[x]$. Then there exist $A,B\in K[x]$ such
that $AF+BG=1$. Then $A(u)F(u)+B(u)G(u)=1$ and hence

(8)
$\displaystyle\min\\{v_{z}^{+}(F(u)),v_{z}^{+}(G(u))\\}\leq\max\\{v_{z}^{-}(A(u)),v_{z}^{-}(B(u))\\}.$

We note that $F(u)$ and $G(u)$ are not indentical zero under the assumption
that $u$ is not algebraic over $K$. Since $u$ has no pole, the right hand side
of (8) is bounded by the number of poles of the coefficients of $A$ and $B$.
By Theorem 2.1, $N_{\beta}(\infty,r)\leq T_{\beta}(r)+O(1)={\rm o}(T_{u}(r))$
for any $\beta\in K$. Therefore, (8) implies that

$\displaystyle N_{\gcd}(F(u),G(u),r)\leq{\rm o}(T_{u}(r)).$

###  2.3. Basic Results

######  Lemma 2.9.

Let ${\mathbf{f}}:\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a
holomorphic map. Then $a^{\prime}\in K_{\mathbf{f}}$ for every $a\in
K_{\mathbf{f}}$.

###### Proof.

We may suppose that $a$ is non-constant. Noting that
$N_{a^{\prime}/a}(\infty,r)\leq N^{(1)}_{a}(0,r)+N^{(1)}_{a}(\infty,r)$,
Theorem 2.1 implies $N_{a^{\prime}/a}(\infty,r)\leq 2T_{a}(r)+O(1)$. On the
other hand, we have from Lemma 2.3 that
$m_{a^{\prime}/a}(\infty,r)\leq_{\operatorname{exc}}O(\log T_{a}(r))$. Now the
condition $a\in K_{\mathbf{f}}$ implies that
$T_{a^{\prime}/a}(r)=o(T_{\mathbf{f}}(r))$ and thus $T_{a^{\prime}}(r)\leq
T_{a^{\prime}/a}(r)+T_{a}(r)=o(T_{\mathbf{f}}(r))$ as desired. ∎

We refer to [9] for proofs of the following statements.

######  Lemma 2.10 ([9, Lemma 2.6]).

Let $f$ be a nonconstant meromorphic function satisfying

$\displaystyle N^{(1)}_{f}(0,r)+N^{(1)}_{f}(\infty,r)={\rm o}(T_{f}(r)).$

Then $T_{f^{\prime}/f}(r)\leq_{\operatorname{exc}}{\rm o}(T_{f}(r))$.

######  Proposition 2.11 ([9, Corollary 2.8]).

Let $u_{0},\dots,u_{n}$ be units, i.e. entire functions without zeros, and let
${\bf u}=(u_{0},\ldots,u_{n})$. Let $K$ be a subfield of $K_{\bf u}$. If
$u_{0},\dots,u_{n}$ are algebraically dependent over $K$, then they are
multiplicatively dependent modulo $K_{\bf u}$.

######  Proposition 2.12 ([9, Corollary 2.10]).

Let $n\geq 2$ and $Q_{1},\dots,Q_{n}\in\mathbb{C}[z].$ If
$e^{Q_{1}},\dots,e^{Q_{n}}$ are algebraically dependent over $\mathbb{C}(z)$,
then they are multiplicatively dependent modulo $\mathbb{C}$.

##  3\. Main Lemmas

Let $u_{1},...,u_{n}$ be nonconstant units, i.e. entire functions without
zeros. For convenience of discussions in the affine situation, we simply
denote by $\mathbf{u}=(u_{1},\ldots,u_{n})$ and

(9) $K_{\mathbf{u}}:=\\{a:a\text{ is a meromorphic function with
}T_{a}(r)=o(\max_{1\leq i\leq n}T_{u_{i}}(r))\\}.$

Let $\mathbf{x}:=(x_{1},\ldots,x_{n})$. For
$\mathbf{i}=(i_{1},\ldots,i_{n})\in\mathbb{Z}^{n}$, we let
$\mathbf{x^{i}}:=x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$ and
$\mathbf{u^{i}}:=u_{1}^{i_{1}}\cdots u_{n}^{i_{n}}$. For a polynomial
$F(\mathbf{x})=\sum_{\mathbf{i}}a_{\mathbf{i}}\mathbf{x}^{\mathbf{i}}\in
K_{\mathbf{u}}[\mathbf{x}]:=K_{\mathbf{u}}[x_{1},\dots,x_{n}]$, we define

(10) $\displaystyle
D_{\mathbf{u}}(F)(\mathbf{x}):=\sum_{\mathbf{i}}\frac{(a_{\mathbf{i}}\mathbf{u}^{\mathbf{i}})^{\prime}}{\mathbf{u}^{\mathbf{i}}}\mathbf{x}^{\mathbf{i}}=\sum_{\mathbf{i}}(a^{\prime}_{\mathbf{i}}+a_{\mathbf{i}}\cdot\sum_{j=1}^{n}i_{j}\frac{u_{j}^{\prime}}{u_{j}})\mathbf{x}^{\mathbf{i}}.$

We note that $D_{\mathbf{u}}(F)(\mathbf{x})\in K_{\mathbf{u}}[\mathbf{x}]$
since $u_{i}^{\prime}/u_{i}\in K_{\mathbf{u}}$, $1\leq i\leq n$, by Lemma
2.10. Furthermore, a direct computation shows that

(11) $\displaystyle F(\mathbf{u})^{\prime}=D_{\mathbf{u}}(F)(\mathbf{u}),$

and the following product rule:

(12) $\displaystyle D_{\mathbf{u}}(FG)=D_{\mathbf{u}}(F)G+FD_{\mathbf{u}}(G)$

for $F,G\in K_{\mathbf{u}}[\mathbf{x}]$. We recall the following from [9].

######  Lemma 3.1 ([9, Lemma 3.1]).

Let $u_{1},...,u_{n}$ be nonconstant entire functions without zeros and
$\mathbf{u}=(u_{1},\ldots,u_{n})$. Let $K$ be a subfield of $K_{\mathbf{u}}$
such that $u_{j}^{\prime}/u_{j}\in K$, $1\leq j\leq n$, and $a^{\prime}\in K$
for any $a\in K$. Let $F$ be a nonconstant polynomial in
$K[x_{1},\dots,x_{n}]$ without monomial factors and let $F=F_{1}^{d_{1}}\cdots
F_{k}^{d_{k}}$, where $k\geq 1$ and $F_{1},\ldots,F_{k}\in
K[x_{1},\dots,x_{n}]$ are distinct irreducible factors of $F$. Suppose that
$u_{1},\dots,u_{n}$ are multiplicatively independent modulo $K$. Then the
following two polynomials $\bar{F}:=F_{1}\cdots F_{k}$, and
$\hat{F}_{\mathbf{u}}:=d_{1}D_{\mathbf{u}}(F_{1})F_{2}\cdots
F_{k}+\cdots+d_{k}F_{1}\cdots F_{k-1}D_{\mathbf{u}}(F_{k})$ are coprime in
$K[x_{1},\dots,x_{n}]$.

The following lemma is a reformulation of [9, Lemma 3.2] with $\alpha$ being
an entire function.

######  Lemma 3.2.

Let $u_{1},...,u_{n}$ be nonconstant entire functions without zeros and
$\mathbf{u}=(u_{1},\ldots,u_{n})$. Let $K$ be a subfield of $K_{\mathbf{u}}$
such that $u_{j}^{\prime}/u_{j}\in K$, $1\leq j\leq n$, and $a^{\prime}\in K$
for any $a\in K$. Let $d\geq 2$ be an integer. Let $F\in K[x_{1},\dots,x_{n}]$
and assume that $F$ has no nonmonomial repeated factors. Assume that
$u_{1},\dots,u_{n}$ are algebraically independent over $K$. Let
$\varepsilon>0$. If $F(\mathbf{u})=\alpha g^{d}$ for some nonzero entire
functions $g$ and $\alpha$, then

$N_{g}(0,r)\leq_{\operatorname{exc}}\varepsilon\max_{1\leq j\leq
n}\\{T_{u_{j}}(r)\\}.$

###### Proof.

Suppose that $F(\mathbf{u})=\alpha g^{d}$ for some nonzero entire functions
$g$ and $\alpha$. If $F=c\in K$, then $N_{g}(0,r)\leq N_{c}(0,r)=o(\max_{1\leq
j\leq n}T_{u_{j}}(r))$ and thus the desired conclusion holds. Hence, we may
assume $F\in K[x_{1},\dots,x_{n}]$ is nonconstant. Since $u_{1},...,u_{n}$ are
entire functions without zeros, we may remove the monomial factors from $F$ if
necessary without changing our assertion. Therefore we may further assume that
$F$ has no monomial factors.

Let $F=F_{1}\cdots F_{k}$, where $F_{1},\ldots,F_{k}\in K[x_{1},\dots,x_{n}]$
are distinct irreducible factors of $F$.

By (11), we have

(13) $\displaystyle g^{d-1}(d\alpha
g^{\prime}+\alpha^{\prime}g)=D_{\mathbf{u}}(F)(\mathbf{u}).$

Since $F$ and $D_{\mathbf{u}}(F)$ are coprime in $K[x_{1},\dots,x_{n}]$ by
Lemma 3.1, for a given $\varepsilon^{\prime}>0$, Theorem 2.8 gives

$\begin{split}N_{\gcd}(F(\mathbf{u}),D_{\mathbf{u}}(F)(\mathbf{u}),r)\leq_{\operatorname{exc}}\varepsilon^{\prime}\max_{1\leq
j\leq n}\\{T_{u_{j}}(r)\\}.\end{split}$

On the other hand, (13) implies that

(14) $N_{\gcd}(F(\mathbf{u}),D_{\mathbf{u}}(F)(\mathbf{u}),r)=N_{\gcd}(\alpha
g^{d},g^{d-1}(d\alpha g^{\prime}+\alpha^{\prime}g),r)\geq N_{g^{d-1}}(0,r),$

since $\alpha$ and $g$ are entire functions. Consequently,

$N_{g}(0,r)\leq_{\operatorname{exc}}\varepsilon^{\prime}\max_{1\leq j\leq
n}\\{T_{u_{j}}(r)\\}.$

∎

##  4\. Proofs of Theorem 1.2

We first show the following.

######  Theorem 4.1.

Let $u_{0},u_{1},\dots,u_{n}$ be nonconstant entire functions without zeros,
i.e. $\mathbf{u}=(u_{0},\ldots,u_{n}):\mathbb{C}\to\mathbb{G}_{m}^{n+1}$. Let
$K$ be a subfield of the small field $K_{\mathbf{u}}$ w.r.t. $\mathbf{u}$ such
that $u_{j}^{\prime}/u_{j}\in K$, $0\leq j\leq n$, and $a^{\prime}\in K$ for
any $a\in K$. Let $d\geq 2$ be an integer. Let $G$ be a nonconstant
homogeneous polynomials in $K[x_{0},\ldots,x_{n}]$ with no repeated
nonmonomial factors in $K[x_{0},\ldots,x_{n}]$.

If $u_{0},\dots,u_{n}$ are algebraically independent over $K$, then for any
$\epsilon>0$

(15) $\displaystyle
N_{G(\mathbf{u})}(0,r)-N^{(1)}_{G(\mathbf{u})}(0,r)\leq\epsilon
T_{\mathbf{u}}(r),$

where $\mathbf{u}:=(u_{0},u_{1},\ldots,u_{n})$.

If we assume furthermore that there is a $z_{0}\in\mathbb{C}$ such that all
the coefficients of $G$ are holomorphic at $z_{0}$ and none of the functions
$G(1,0,\dots,0),\dots,$ $G(0,\dots,0,1)$ vanishes at $z_{0}$, then

$\displaystyle N^{(1)}_{G(\mathbf{u})}(0,r)\geq_{\rm exc}(\deg
G-\epsilon)\cdot T_{\mathbf{u}}(r).$

###### Proof.

Let $z_{0}\in\mathbb{C}$. If $v_{z_{0}}(G(\mathbf{u}))\geq 2$, then it follows
from (11) that
$v_{z_{0}}(D_{\mathbf{u}}(G)(\mathbf{u}))=v_{z_{0}}(G(\mathbf{u}))-1$. Hence,

$\min\\{v_{z_{0}}^{+}(G(\mathbf{u})),v_{z_{0}}^{+}(D_{\mathbf{u}}(G)(\mathbf{u}))\\}\geq
v_{z_{0}}^{+}(G(\mathbf{u}))-\min\\{1,v_{z_{0}}^{+}(G(\mathbf{u}))\\}.$

Consequently,

(16) $\displaystyle
N_{\gcd}(G(\mathbf{u}),D_{\mathbf{u}}(G)(\mathbf{u}),r)\geq
N_{G(\mathbf{u})}(0,r)-N^{(1)}_{G(\mathbf{u})}(0,r).$

By Lemma 3.1, $G$ and $D_{\mathbf{u}}(G)$ are comprime and hence we can apply
Theorem 2.8 to get

(17) $\displaystyle
N_{\gcd}(G(\mathbf{u}),D_{\mathbf{u}}(G)(\mathbf{u}),r)\leq_{\rm
exc}\frac{\epsilon}{2}T_{\mathbf{u}}(r)$

for any $\epsilon>0$. Then we have

(18) $\displaystyle
N_{G(\mathbf{u})}(0,r)-N^{(1)}_{G(\mathbf{u})}(0,r)\leq_{\operatorname{exc}}\frac{\epsilon}{2}T_{\mathbf{u}}(r).$

This proves the first assertion.

On the other hand, the assumption that $u_{0},\dots,u_{n}$ are algebraically
independent over $K$ together with the existence of $z_{0}$ such that all the
coefficients of $G$ are holomorphic at $z_{0}$ and none of the
$G(1,0,\dots,0),\dots,G(0,\dots,0,1)$ vanishes at $z_{0}$ allows us to use
Theorem 2.6 with polynomials $G$, $x_{0},\ldots,x_{n}$ to obtain

(19) $\displaystyle N_{G(\mathbf{u})}(0,r)\geq_{\rm exc}\deg
G(1-\frac{\epsilon}{2\deg G})T_{\mathbf{u}}(r)=(\deg
G-\frac{\epsilon}{2})T_{\mathbf{u}}(r),$

as $u_{i}$ is entire without zero for every $0\leq i\leq n$. Combining (19)
with (18), we obtain the second assertion. ∎

###### of Theorem 1.2.

By Lemma 2.10, $u_{j}^{\prime}/u_{j}\in K_{\mathbf{u}}$, $1\leq j\leq n$, and
by Lemma 2.9 the condition $a^{\prime}\in K_{\mathbf{u}}$ for any $a\in
K_{\mathbf{u}}$ also holds. Therefore, the first assertion follows from
applying both Proposition 2.11 and Theorem 4.1 with $K=K_{\mathbf{u}}$. To
deduce the second assertion from Theorem 4.1 applied with $K=K_{\mathbf{u}}$,
we only have to note that $G(1,0,\dots,0),\dots,G(0,\dots,0,1)$ and the
coefficients of $G$ are meromorphic functions and thus there is a
$z_{0}\in\mathbb{C}$ such that all the coefficients of $G$ are holomorphic at
$z_{0}$ and none of the $G(1,0,\dots,0),\dots,G(0,\dots,0,1)$ vanishes at
$z_{0}$ since we assume that none of them is identically zero. Finally, we
note that the assumption that $u_{0},\dots,u_{n}$ are entire functions with no
zeros of finite order implies that $u_{j}^{\prime}/u_{j}\in\mathbb{C}(z)$,
$1\leq j\leq n$, and that $a^{\prime}\in\mathbb{C}(z)$ for any
$a\in\mathbb{C}(z)$; also, by Proposition 2.12, the assumption that
$u_{0},\dots,u_{n}$ are multiplicatively independent modulo $\mathbb{C}$
implies that $u_{0},\dots,u_{n}$ multiplicatively independent modulo
$\mathbb{C}(z)$; Therefore the third assertion follows from applying previous
arguments with $K=\mathbb{C}(z)$. ∎

##  5\. Proofs of Theorem 1.3

###### of Theorem 1.3.

Letting $K\subset K_{\mathbf{f}}$ be a subfield, we consider the following
arguments.

Let $z_{0}\in\mathbb{C}$ such that all the coefficients of all $F_{i}$, $1\leq
i\leq n+1$ are holomorphic at $z_{0}$ and the zero locus of $F_{i}$, $1\leq
i\leq n+1$, evaluating at $z_{0}$ intersect transversally. We note these
conditions imply that $z_{0}$ is not a common zero of the coefficients of
$F_{i}$, for each $1\leq i\leq n+1$.

For each polynomial $G=\sum_{I}a_{I}{{\bf x}^{I}}\in K[x_{0},\ldots,x_{n}]$,
where $I=(i_{0},\ldots,i_{n})\in\mathbb{Z}_{\geq 0}^{n+1}$ and ${\bf
x}^{I}=x_{0}^{i_{0}}\cdots x_{n}^{i_{n}}$, we denote by
$G({z_{0}}):=\sum_{I}a_{I}(z_{0}){{\bf x}^{I}}$ if all the coefficients of $G$
are holomorphic at $z_{0}$ and do not vanish simultaneously at $z_{0}$. Denote
by $D_{i}\subset\mathbb{P}^{n}(\overline{K})$ the divisor (over $K$) defined
by $F_{i}$, by $D_{i}(z_{0})\subset\mathbb{P}^{n}(\mathbb{C})$ the divisor
defined by $F_{i}(z_{0})$, and let $D\coloneqq D_{1}+\cdots+D_{n+1}$. Since
$D_{i}(z_{0})$, $1\leq i\leq n+1$, intersect transversally, they are in
general position; thus the set of polynomials $F_{i}$, $1\leq i\leq n+1$, is
in weakly general position. Then Proposition 2.7 implies that the only
$(x_{0},\ldots,x_{n})\in\overline{K}^{n+1}$ with $F_{i}(x_{0},\ldots,x_{n})=0$
for each $1\leq i\leq n+1$ is $(0,\ldots,0)$. Thus the association
$P\mapsto[F_{1}^{a_{1}}(P):\ldots:F_{n+1}^{a_{n+1}}(P)]$, where
$a_{i}\coloneqq{\rm lcm}(\deg F_{1},\ldots,\deg F_{n+1})/\deg F_{i}$, defines
a morphism $\pi:\mathbb{P}^{n}(\overline{K})\to\mathbb{P}^{n}(\overline{K})$
over $K$.

It is well-known that the ramification divisor of $\pi$ is the zero locus of
the determinant $J\in K[x_{0},\ldots,x_{n}]$ of the Jacobian matrix

$\big{(}\frac{\partial F_{i}^{a_{i}}}{\partial x_{j}}\big{)}_{1\leq i\leq
n+1,0\leq j\leq n}$

of $\pi$. Our plan is to show that there exists an irreducible factor
$\tilde{G}$ of $J$ in $K_{\mathbf{f}}[x_{0},\ldots,x_{n}]$ such that the
corresponding moving hypersurfaces of $\tilde{G}$, $F_{1},\ldots,F_{n+1}$ are
in weakly general position. Furthermore, we will show that
$\tilde{G}(\mathbf{f})$ has very few zeros and hence conclude that
$\mathbf{f}$ is algebraically degenerate by applying Theorem 2.6 for
$\tilde{G}$, $F_{1},\ldots,F_{n+1}$.

For this purpose, we look at the specialization of $J$ at $z_{0}$. Denote by
$\pi|_{z_{0}}=[F_{1}^{a_{1}}(z_{0}):\ldots:F_{n+1}^{a_{n+1}}(z_{0})]:\mathbb{P}^{n}(\mathbb{C})\rightarrow\mathbb{P}^{n}(\mathbb{C})$,
which is a morphism since $F_{1}(z_{0}),\ldots,F_{n+1}(z_{0})$ are in general
position. Then $J(z_{0})\in\mathbb{C}[x_{0},\ldots,x_{n}]$ is the determinant
of the Jacobian matrix of $\pi|_{z_{0}}$. Observing that $J$ has a factor $G$
which denotes the determinant of

$\displaystyle M:=\big{(}\frac{\partial F_{i}}{\partial x_{j}}\big{)}_{1\leq
i\leq n+1,0\leq j\leq n}$

in ${\color[rgb]{0,0,1}K}[x_{0},\ldots,x_{n}]$, we see that $G(z_{0})$ is a
factor of $J(z_{0})$. We note that $G(z_{0})$ is not a constant since each
$F_{i}$ is homogeneous and reduced and $\sum_{i=1}^{n+1}F_{i}\geq n+2$. We
claim that $[G(z_{0})=0]$ (the zero locus of $G(z_{0})$),
$D_{1}(z_{0}),\ldots,D_{n+1}(z_{0})$ are is in general position (in
$\mathbb{P}^{n}(\mathbb{C})$). To prove this, it suffices to show that
$G(z_{0})$ does not vanish at any intersection point of any $n$ divisors among
$D_{1}(z_{0}),\ldots,D_{n+1}(z_{0})$. By rearranging the index, it suffices to
consider that $P\in\cap_{i=1}^{n}D_{i}(z_{0})$ and show that $G(z_{0})(P)\neq
0$. Since the $D_{i}(z_{0})$’s are in general position, we see that
$F_{n+1}(z_{0})(P)\neq 0$. Using the Euler formula

$\sum_{j=1}^{n}\frac{\partial F_{i}(z_{0})}{\partial x_{j}}x_{j}=\deg
F_{i}(z_{0})\cdot F_{i}(z_{0}),$

we obtain

$\begin{split}x_{0}G(z_{0})=\det\begin{pmatrix}d_{1}F_{1}(z_{0})&\frac{\partial
F_{1}(z_{0})}{\partial x_{1}}&\cdots&\frac{\partial F_{1}(z_{0})}{\partial
x_{n}}\\\ \vdots&\vdots&\vdots&\vdots\\\ d_{n+1}F_{n+1}(z_{0})&\frac{\partial
F_{n+1}(z_{0})}{\partial x_{1}}&\cdots&\frac{\partial F_{n+1}(z_{0})}{\partial
x_{n}}\end{pmatrix},\end{split}$

and hence

$x_{0}(P)(G(z_{0}))(P)=(-1)^{n+1}d_{n+1}(F_{n+1}(z_{0}))(P)\det\left(\frac{\partial
F_{i}(z_{0})}{\partial x_{j}}(P)\right)_{1\leq i,j\leq n},$

where $d_{i}:=\deg F_{i}=\deg(F_{i}(z_{0}))$. Since
$D_{1}(z_{0}),\ldots,D_{n+1}(z_{0})$ intersect transversally, we see that
$\det\left(\frac{\partial F_{i}(z_{0})}{\partial x_{j}}(P)\right)_{1\leq
i,j\leq n}\neq 0$. Then $G(z_{0})(P)\neq 0$ as $F_{n+1}(z_{0})(P)\neq 0$. This
proves our claim; it actually shows that every irreducible factor of
$G(z_{0})$ is in general position with $F_{i}(z_{0})$, $1\leq i\leq n+1$.
Hence, there is a nonconstant irreducible factor $\tilde{G}$ of $G$ (and hence
of $J$) in $K[x_{0},\ldots,x_{n}]$ such that $\tilde{G}$,
$F_{1},\ldots,F_{n+1}$ is in weakly general position.

We note that the finite map
$\pi:\mathbb{P}^{n}(\overline{K})\to\mathbb{P}^{n}(\overline{K})$ is defined
over $K$; by the definition of $D$, it induces a finite morphism
$\tilde{\pi}:=[F_{1}^{a_{1}}/F_{n+1}^{a_{n+1}}:\ldots:F_{n}^{a_{n}}/F_{n+1}^{a_{n+1}}]:\mathbb{P}^{n}\setminus{\rm
Supp}(D)\mapsto\mathbb{G}_{m}^{n}$, which is a morphism between affine
varieties over $K$. Denote by $Y$ the zero locus of $\tilde{G}$ in
$\mathbb{P}^{n}\setminus{\rm Supp}(D)$; by our construction, $Y$ is contained
in the ramification divisor of $\tilde{\pi}$. Then there exists an irreducible
polynomial $A\in K[y_{1},\ldots,y_{n}]$ such that the vanishing order of
$\tilde{\pi}^{*}A$ along $Y$ is at least 2. Let $\tilde{A}$ be the
homogenization of $A$. Then this construction gives
$\pi^{*}\circ\tilde{A}=\tilde{G}^{2}H$ for some $H\in K[x_{0},\ldots,x_{n}]$.
Now let $\mathbf{f}=(f_{0},\ldots,f_{n}):\mathbb{C}\to\mathbb{P}^{n}$ be a
holomorphic map, where $f_{0},\ldots,f_{n}$ are entire functions without
common zeros, such that
$\mathbf{u}\coloneqq\pi(\mathbf{f})=(F_{1}(\mathbf{f})^{a_{1}},\ldots,F_{n+1}(\mathbf{f})^{a_{n+1}})$
is a tuple of entire functions without zeros. From the equality
$\tilde{A}(\mathbf{u})=(\pi^{*}\circ\tilde{A})(\mathbf{f})=\tilde{G}^{2}(\mathbf{f})H(\mathbf{f})$,
it follows that for each $z\in\mathbb{C}$ with
$v_{z}(\tilde{G}(\mathbf{f}))>0$, i.e. $\mathbf{f}(z)\in Y$, we have

(20) $\displaystyle v_{z}(\tilde{A}(\mathbf{u}))\geq
2v_{z}(\tilde{G}(\mathbf{f}))+\min\\{0,v_{z}(H(\mathbf{f}))\\}\geq
v_{z}(\tilde{G}(\mathbf{f}))+1+\min\\{0,v_{z}(H(\mathbf{f}))\\}.$

Since $f_{0},\ldots,f_{n}$ are entire functions, the nonnegative number
$-\min\\{0,v_{z}(H(\mathbf{f}))\\}$ is bounded by the number of poles of the
coefficients of $H$ at $z$. Since $N_{\beta}(\infty,r)\leq
T_{\beta}(r)+O(1)={\rm o}(T_{\mathbf{f}}(r))$ for any $\beta\in K$, it follows
from (20) that

$N_{\tilde{G}(\mathbf{f})}(0,r)\leq
N_{A(\mathbf{u})}(0,r)-N_{A(\mathbf{u})}^{(1)}(0,r)+{\rm
o}(T_{\mathbf{f}}(r)).$

Now consider in the cases where $K=K_{\mathbf{f}}$ and where
$K=\mathbb{C}(z)$. As noticed in the proof of Theorem 1.3, the property
required for $(K,\mathbf{u})$ in Theorem 4.1 is satisfied. By (15) in Theorem
4.1 and that $T_{\mathbf{u}}(r)=O(T_{\mathbf{f}}(r))$, we have

(21) $\displaystyle N_{\tilde{G}(\mathbf{f})}(0,r)\leq_{\rm exc}\epsilon
T_{\mathbf{f}}(r)$

for every $\epsilon>0$.

Finally, since the set of polynomials $\tilde{G}$, $F_{1},\ldots,F_{n+1}$ is
in weakly general position, Theorem 2.6 shows that if $\mathbf{f}$ is
algebraically nondegenerate over $K$, then we reach the contradiction that for
every $\epsilon>0$

$\displaystyle\left(1-\epsilon\right)T_{\mathbf{f}}(r)\leq\frac{1}{\deg\tilde{G}}N_{\tilde{G}(\mathbf{f})}(0,r)+\sum_{j=1}^{n+1}\frac{1}{\deg
F_{j}}N_{F_{j}(\mathbf{f})}(0,r)\leq_{\rm exc}\epsilon T_{\mathbf{f}}(r),$

where the second inequality uses (21) and the hypothesis that each entire
function $F_{i}(\mathbf{f})$ has no zeros. This shows that $\mathbf{f}$ is
algebraically degenerate over $K$, and proves the first part of the desired
conclusions when the case where $K=K_{\mathbf{f}}$ is considered.

It remains to consider when $F_{i}\in\mathbb{C}[x_{0},\ldots,x_{n}]$ and
$\mathbf{f}$ is of finite order. Our previous argument for the case where
$K=\mathbb{C}(z)$ shows that $\mathbf{f}$ is algebraically degenerate over
$\mathbb{C}(z)$. By considering the transcendence degree of the extension
generated by $F_{1}(\mathbf{f}),\ldots,F_{n+1}(\mathbf{f})$ over
$\mathbb{C}(z)$, we see that they are algebraic dependent over
$\mathbb{C}(z)$. Since they are units of finite order, Proposition 2.12
implies that $F_{1}(\mathbf{f}),\ldots,F_{n+1}(\mathbf{f})$ are
multiplicatively dependent modulo $\mathbb{C}$. As the coefficients of all
$F_{i}$ are in $\mathbb{C}$, it implies that $\mathbf{f}$ is algebraically
degenerate over $\mathbb{C}$. ∎

##  6\. Proof of Theorem 1.5

The proof of Theorem 1.5 is an adaption and generalization of the proof of [3,
Theorem 3] to the complex situation. We reformulate and generalize the
arguments in [3] into the following two lemmas to complete the proof.

######  Lemma 6.1.

Let $d$ and $q$ be positive integers. Let
$\mu_{1},\ldots,\mu_{n}\in\mathbb{C}^{*}$ and
$\mathbf{u}=(u_{1},\ldots,u_{n})=(e^{\mu_{1}z^{q_{1}}},\ldots,e^{\mu_{n}z^{q_{n}}})$
with $q=q_{1}=q_{2}=\cdots=q_{\ell}>q_{\ell+1}\geq\cdots\geq q_{n}\geq 1$ for
some $1\leq\ell\leq n$. Assume that $u_{1},\ldots,u_{n}$ are algebraic
independent over $\mathbb{C}(z)$; this means that $\mathbb{C}(z)[u_{1}^{\pm
1},\ldots,u_{n}^{\pm 1}]$ can be regarded as the ring of Laurent polynomials
in variables $u_{1},\ldots,u_{n}$ over $\mathbb{C}(z)$. Let
$A_{i}\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$, $0\leq i\leq
d-1$. Assume that the polynomial
$F(Y):=Y^{d}+A_{d-1}Y^{d-1}+\ldots+A_{1}Y+A_{0}\in\mathbb{C}(z)[u_{1}^{\pm
1},\ldots,u_{n}^{\pm 1}][Y]$ is irreducible and its discriminant
$\Delta\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$ is square-free.
Suppose that there exists an entire function $g$ such that $F(g)=0$. Then
there exists $Q\in\mathbb{C}(z)][u_{\ell+1}^{\pm 1},\dots,u_{n}^{\pm 1}]$ such
that

$\displaystyle\Delta=Qu_{1}^{m_{1}}\cdots u_{\ell}^{m_{\ell}},$

where $m_{i}\in\mathbb{Z}$ for $1\leq i\leq\ell$.

The proof of Lemma 6.1 will be given at the end of this section.

######  Lemma 6.2.

Let $k$ be a field of characteristic zero. Suppose that $f(Y)\in k[U_{1}^{\pm
1},\ldots,U_{n}^{\pm 1}][Y]$ is a monic and irreducible polynomial in $Y$ such
that its discriminant $\Delta\in k[U_{1}^{\pm 1},\ldots,U_{n}^{\pm 1}]$ (with
respect to $Y$) euqals $QU_{1}^{m}$ for some $Q\in k[U_{2}^{\pm
1},\ldots,U_{n}^{\pm 1}]$ and $m\in\mathbb{Z}$. Then we have

$\displaystyle
f(U_{1},\ldots,U_{n},Y)=U_{1}^{s}P(U_{2},\ldots,U_{n},U_{1}^{t}Y+A(U_{1},\ldots,U_{n})),$

where $P\in k[U_{2}^{\pm 1},\ldots,U_{n}^{\pm 1},W]$,
$A(U_{1},\ldots,U_{n})\in k[U_{1}^{\pm 1},\ldots,U_{n}^{\pm 1}]$ and
$s,t\in\mathbb{Z}$.

We note that Lemma 6.2 is verified in the middle of the proof of [3, Theorem
3] for $n=2$ (See [3, Eq. (12)].) based on the following claim: Let
$\mathbf{K}$ be an algebraically closed field of characteristic 0 and let
$F\in\mathbf{K}[V,Y]$ be an irreducible polynomial monic and of degree $e$ in
$Y$ such that its discriminant with respect to $Y$ is a constant times a power
of $V$. Then $g(V,Y)=(Y-a(V))^{e}-bV^{s}$ where $a\in\mathbf{K}[V]$,
$b\in\mathbf{K}^{*}$ and $s$ is an integer prime to $e$. It is clear that
their arguments extend naturally to all positive integer $n$ by taking
$\mathbf{K}$ to be an algebraic closure of $k$ if $n=1$; and an algebraic
closure of $k(U_{2},\ldots,U_{n})$ if $n\geq 2$.  We will not repeat the proof
here.

###### of Theorem 1.5.

We prove the desired conclusion by induction on $n$. In the case where $n=0$,
i.e. each $A_{i}$ is in $\mathbb{C}(z)$, the assumption $F(g)=0$ says that the
entire function $g$ is algebraic over $\mathbb{C}(z)$ and hence
$T_{g}(r)=O(\log r)$. By Theorem 2.2, we conclude that $g\in\mathbb{C}(z)$ and
thus $g\in\mathbb{C}[z]\subset\mathcal{E}_{0}$. This settles down the base
case of the induction.

Suppose that $n>0$. We may rearrange the $u_{i}$ such that

$q=q_{1}=\cdots=q_{i_{1}}>q_{i_{1}+1}=\cdots=q_{i_{2}}>\cdots>q_{i_{b}+1}=\ldots=q_{n}.$

As $u_{1},\ldots,u_{n}$ are multiplicatively independent modulo $\mathbb{C}$,
it follows from Proposition 2.12 that they are algebraically independent over
$\mathbb{C}(z)$. Note that $\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm
1}]\subset\mathcal{K}_{q}$ and thus our hypothesis implies that $F(Y)$ is
irreducible over $\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$. By Lemma
6.1, there exists $Q\in\mathbb{C}(z)[u_{i_{1}+1}^{\pm 1},\dots,u_{n}^{\pm 1}]$
such that

$\displaystyle\Delta=Qu_{1}^{m_{1}}\cdots u_{i_{1}}^{m_{i_{1}}},$

where $m_{i}\in\mathbb{Z}$ for $1\leq i\leq i_{1}$. By Lemma 6.2, we have

(22) $\displaystyle F(Y)=u_{1}^{s_{1}}P(u_{1}^{t_{1}}Y-c),$

for some $s_{1}\in\mathbb{Z}$, $P\in\mathbb{C}(z)[u_{2}^{\pm
1},\ldots,u_{n}^{\pm 1}][W]$, $t_{1}\in\mathbb{Z}$ and
$c\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}]$. From (22), we see
that the leading coefficient of $P(W)$ in $W$ must be a power of $u_{1}$ since
$F(Y)\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1}][Y]$ is monic in $Y$.
But since the coefficients of $P$ are in $\mathbb{C}(z)[u_{2}^{\pm
1},\ldots,u_{n}^{\pm 1}]$, it follows that $P$ is monic in $W$. Similarly, we
see that the discriminant of $P$ with respect to $W$ is
$\Delta_{P}=Qu_{2}^{m_{2}}\cdots u_{i_{1}}^{m_{i_{1}}}$, which is square-free.
Note also that $P$ is irreducible in $W$ over $\mathbb{C}(z)[u_{2}^{\pm
1},\ldots,u_{n}^{\pm 1}]$ since $F(Y)$ is irreducible in $Y$. Now (22) gives
$P(u_{1}^{t_{1}}g-c)=u_{1}^{-s_{1}}F(g)=0$, which implies that
$u_{1}^{t_{1}}g-c\in\mathcal{E}_{q}$ by induction hypothesis; since
$c\in\mathcal{E}_{q}$ and $u_{1}\in\mathcal{E}_{q}^{*}$, we conclude
$g\in\mathcal{E}_{q}$ as desired. ∎

The strategy of proving Lemma 6.1 is motivated by the first part of the proof
of [3, Theorem 3]. We will first show that thenumber of zeros of
$F^{\prime}(g)$ is “small”. Then we can use Theorem 2.5, a truncated second
main theorem, to the algebraic relation of $F^{\prime}(g)$ and
$u_{1},\ldots,u_{n}$ to conclude the assertion on that the discriminant
$\Delta$.

###### of Lemma 6.1.

The assertion is clear when $d=1$. Therefore, we assume that $d\geq 2$. Denote
by $\alpha_{1},\ldots,\alpha_{d}$ all the roots of $F$ in a fixed algebraic
closure of $\mathbb{C}(z,u_{1},\ldots,u_{n})$ such that $\alpha_{1}=g$ is an
entire function. Then we have $\Delta=\prod_{i<j}(\alpha_{i}-\alpha_{j})^{2}$,
which is not zero since $F$ is irreducible.

Let $H:=F(Y)/(Y-g)=\prod_{1<i\leq d}(Y-\alpha_{i})$. The division algorithm
shows that $H\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1},g][Y]$.
Letting $G\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm 1},g]$ be the
discriminant of $H$ with respect to $Y$, we note that the entire function $G$
is equal to $\prod_{1<i<j\leq d}(\alpha_{i}-\alpha_{j})^{2}$, which is
nonzero.

Note that $F^{\prime}(g)=\prod_{i>1}(g-\alpha_{i})$ is a nonzero entire
function, where $F^{\prime}$ denotes the formal derivative of $F$ with respect
to $Y$. Hence

(23) $\displaystyle\Delta=F^{\prime}(g)^{2}G.$

Let $\epsilon_{1}$ be a sufficiently small positive number. We claim that

(24) $\displaystyle N_{F^{\prime}(g)}{\color[rgb]{0,0,1}(0,}r)\leq_{{\rm
exc}}\epsilon_{1}\cdot r^{q}+o(r^{q}).$

By our assumption that $\Delta\in\mathbb{C}(z)[u_{1}^{\pm 1},\ldots,u_{n}^{\pm
1}]$ is square-free, we may write

(25) $\displaystyle\Delta=u_{1}^{m_{1}}\cdots
u_{n}^{m_{n}}A(u_{1},\ldots,u_{n})$

with integers $m_{1},\ldots,m_{n}$ and a square-free polynomial
$A\in\mathbb{C}(z)[x_{1},\ldots,x_{n}]$. Since $u_{i}$, $1\leq i\leq n$, are
entire functions without zeros, we have from (23) and (25) that there exists a
non-zero entire function $\gamma$ such that

(26) $\displaystyle A(u_{1},\ldots,u_{n})=F^{\prime}(g)^{2}\gamma.$

We note that elements in $\mathbb{C}(z)$ have slow growth with respect to
$\mathbf{u}$. As $u_{1},\ldots,u_{n}$ are algebraically independent over
$\mathbb{C}(z)$, we have (24) from Lemma 3.2.

Since $F(g)=0$ with $F\in\mathbb{C}(z,u_{1},\ldots,u_{n})[Y]$, it follows that
$g$ is algebraic over $\mathbb{C}(z,u_{1},\ldots,u_{n})$ and that
$F^{\prime}\in\mathbb{C}(z,u_{1},\ldots,u_{n})[Y]$, so $F^{\prime}(g)$ is also
algebraic over $\mathbb{C}(z,u_{1},\ldots,u_{n})$. Hence there exists a
polynomial $R\in\mathbb{C}[z][U_{1},\ldots,U_{n},Y]$ of the fewest terms such
that

(27) $\displaystyle R(u_{1},\ldots,u_{n},F^{\prime}(g))=0.$

Since $F^{\prime}(g)\neq 0$, we may assume that with respect to the variable
$Y$, the polynomial $R$ has a nonzero constant term. Note that (27) can be
written as

(28)
$\displaystyle\sum_{I=(i_{0},\ldots,i_{n})\in\Sigma}c_{I}F^{\prime}(g)^{i_{0}}u_{1}^{i_{1}}\cdots
u_{n}^{i_{n}}=0,$

where $\Sigma$ is the set of vectors $(i_{0},\ldots,i_{n})$ corresponding to
monomials appearing in $R$ and $c_{I}\in\mathbb{C}[z]$, not identically zero.
By construction, no proper subsum of the left hand side of (28) vanishes, and
$\Sigma$ must contain a vector $I_{0}=(0,a_{1},\ldots,a_{n})$ starting with
$0$.

Note that in the left hand side of (28), the nonzero term corresponding to
$I_{0}$ has only finitely many zeros. Thus there is some $h\in\mathbb{C}[z]$
such that all terms in the left hand side of the following equality equivalent
to (28)

(29)
$\displaystyle\sum_{I=(i_{0},\ldots,i_{n})\in\Sigma}h^{-1}c_{I}F^{\prime}(g)^{i_{0}}u_{1}^{i_{1}}\cdots
u_{n}^{i_{n}}=0$

are entire functions without common zeros.

Let $\epsilon$ be a sufficiently small positive real number. Applying Theorem
2.5 to (29), with the estimate in (24) and Lemma 2.4, we have

(30) $\displaystyle T_{F^{\prime}(g)^{i_{0}}u_{1}^{i_{1}-a_{1}}\cdots
u_{n}^{i_{n}-a_{n}}}(r)\leq_{{\rm exc}}\epsilon\cdot r^{q}+{\rm o}(r^{q})$

for each $(i_{0},\ldots,i_{n})\in\Sigma$.

On the other hand, the assumption that $u_{1},\ldots,u_{n}$ are
multiplicatively independent modulo $\mathbb{C}$ implies that if the index
vector $J=(j_{1},\ldots,j_{\ell})\neq(0,\ldots,0)$, then

(31) $\displaystyle T_{u_{1}^{j_{1}}\cdots
u_{n}^{j_{n}}}(r)=\gamma_{J}r^{q}+o(r^{q})$

for some positive real $\gamma_{J}$. We first notice that (30) implies that if
$(a_{1},\cdots,a_{\ell})\neq(0,\ldots,0)$, then the first $\ell+1$ coordinates
of any other vector in $\Sigma$ with $i_{0}=0$ must be equal to
$(0,a_{1},\cdots,a_{\ell})$. Let $M_{1}=U_{1}^{a_{1}}\cdots
U_{\ell}^{a_{\ell}}$ and $\tilde{R}=M_{1}^{-1}R.$ We have
${\tilde{R}}(u_{1},\ldots,u_{n},F^{\prime}(g))=\sum_{I=(i_{0},\ldots,i_{n})\in\Sigma^{\prime}}c_{I}F^{\prime}(g)^{i_{0}}u_{1}^{i_{1}}\cdots
u_{n}^{i_{n}}$, where
$\Sigma^{\prime}:=\\{(i_{0},i_{1}-a_{1},\ldots,i_{\ell}-a_{\ell},i_{\ell+1}\ldots,i_{n}):(i_{0},\ldots,i_{n})\in\Sigma\\}$.
Note that if $(0,i_{1}\ldots,i_{n})\in\Sigma^{\prime}$, then
$i_{1}=\cdots=i_{\ell}=0$. Since
${\tilde{R}}(u_{1},\ldots,u_{n},F^{\prime}(g))=0$, it follows for any
$(i_{0},i_{1},\ldots,i_{n}),(j_{0},j_{1},\ldots,j_{n})\in\Sigma^{\prime}$ that

$\displaystyle T_{u_{1}^{i_{1}j_{0}-i_{0}j_{1}}\cdots
u_{n}^{i_{n}j_{0}-i_{0}j_{n}}}(r)$ $\displaystyle\leq$ $\displaystyle
j_{0}T_{F^{\prime}(g)^{i_{0}}u_{1}^{i_{1}}\cdots
u_{n}^{i_{n}}}(r)+i_{0}T_{F^{\prime}(g)^{j_{0}}u_{1}^{j_{1}}\cdots
u_{n}^{j_{n}}}(r)$ $\displaystyle\leq$ $\displaystyle\epsilon r^{q}+{\rm
o}(r^{q}),$

where the second inequality is obtained as (30) is; then (31) implies that
$j_{0}(i_{0},i_{1},\cdots,i_{\ell})=i_{0}(j_{0},j_{1},\cdots,j_{\ell})$. Then
there is some tuple $(c_{0},\ldots,c_{\ell})$ of integers with $c_{0}\geq 0$
such that for every $I=(i_{0},i_{1},\ldots,i_{n})\in\Sigma^{\prime}$ we have
that $(i_{0},i_{1},\cdots,i_{\ell})=e_{I}(c_{0},\ldots,c_{\ell})$ for some
nonnegative integer $e_{I}$. Hence we see that

(32) $\displaystyle
0=\tilde{R}(u_{1},\ldots,u_{n},F^{\prime}(g))=R_{1}(u_{1}^{c_{1}}\ldots
u_{\ell}^{c_{\ell}}F^{\prime}(g)^{c_{0}})$

with some $R_{1}\in\mathbb{C}[z,u_{\ell+1},\dots,u_{n}][X]$.

It follows immediately from (32) that $u_{1}^{c_{1}}\ldots
u_{\ell}^{c_{\ell}}F^{\prime}(g)^{c_{0}}$ is algebraic over
$\mathbb{C}(z)(u_{\ell+1},\ldots,u_{n})$. Letting $u:=u_{1}^{c_{1}}\ldots
u_{\ell}^{c_{\ell}}$ and recalling that $\alpha_{2},\ldots,\alpha_{n}$ are all
the conjugates of $g=\alpha_{1}$ over $\mathbb{C}(z)(u_{1},\ldots,u_{n})$, we
see that $uF^{\prime}(\alpha_{i})^{c_{0}}$ is algebraic over
$\mathbb{C}(z)(u_{\ell+1},\ldots,u_{n})$ for all $i=1,\ldots,d$. Since
$\Delta=\prod_{i=1}^{d}F^{\prime}(\alpha_{i})$, we deduce that
$u^{d}\Delta^{c_{0}}$ is algebraic over
$\mathbb{C}(z)(u_{\ell+1},\dots,u_{n})$. For $j=1,\ldots,\ell$, by considering
the minimal polynomial of $u^{d}\Delta^{c_{0}}$ over
$\mathbb{C}(z)(u_{\ell+1},\dots,u_{n})$, we find that the differentiation of
$u^{d}\Delta^{c_{0}}$ with respect to the transcendental element $u_{j}$ over
$\mathbb{C}(z)$ is zero. Hence we conclude that
$u^{d}\Delta^{c_{0}}\in\mathbb{C}[z][u_{\ell+1}^{\pm 1},\dots,u_{n}^{\pm 1}]$
and thus the desired conclusion follows. ∎

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