††footnotetext:  2010 Mathematics Subject Classification: Primary 32H30,
32A22; Secondary 30D35.
Key words and phrases: Kähler manifold, uniqueness theorem, meromorphic
mapping, hyperplane.

# Meromorphic mappings on Kähler manifolds weakly sharing hyperplanes in
${\mathbb{P}}^{n}({\mathbb{C}})$

Si Duc Quang

###### Abstract.

In this paper, we study the uniqueness problem for linearly nondegenerate
meromorphic mappings from a Kähler manifold into
${\mathbb{P}}^{n}({\mathbb{C}})$ satisfying a condition $(C_{\rho})$ and
sharing hyperplanes in general position, where the condition that two
meromorphic mappings $f,g$ have the same inverse image for some hyperplanes
$H$ is replaced by a weaker one that $f^{-1}(H)\subset g^{-1}(H)$. Moreover,
we also give some improvements on the uniqueness problem and algebraic
dependence problem of meromorphic mappings which share hyperplanes and satisfy
$(C_{\rho})$ conditions for different non-negative numbers $\rho$.

##  1\. Introduction

Let $f:M\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a meromorphic mapping of
an $m$-dimensional complete connected Kähler manifold $M$, whose universal
covering is biholomorphic to a ball
$\mathbb{B}(R_{0})=\\{z\in{\mathbb{C}}^{m};\|z\|<R_{0}\\}\
(0<R_{0}\leq+\infty)$, into ${\mathbb{P}}^{n}({\mathbb{C}})$. For $\rho\geq
0$, we say that $f$ satisfies the condition $(C_{\rho})$ if there exists a
nonzero bounded continuous real-valued function $h$ on $M$ such that

$\rho\Omega_{f}+{\rm dd}^{c}\log h^{2}\geq\operatorname{Ric}\omega,$

where $\Omega_{f}$ is the full-back of the Fubini-Study form $\Omega$ on
${\mathbb{P}}^{n}({\mathbb{C}}),\omega=\frac{\sqrt{-1}}{2}\sum_{i,j}h_{i\bar{j}}dz_{i}\wedge$
$d\bar{z}_{j}$ is the Kähler form on
$M,\operatorname{Ric}\omega=dd^{c}\log\left(\operatorname{det}\left(h_{i\bar{j}}\right)\right),d=\partial+\bar{\partial}$
and $d^{c}=\frac{\sqrt{-1}}{4\pi}(\bar{\partial}-$ $\partial)$.

Take a local holomorphic coordinates $(U,z)$ of $M$, where
$z=(z_{1},\ldots,z_{n})$ and $U$ is a Cousin II domain. Let
$f=(f_{0}:\cdots:f_{n})$ be a (local) reduced representation of $f$ on $U$.
Denote by $\mathcal{M}_{p}$ the field of all germs of meromorphic functions at
a point $p\in U$. Denote by $\mathcal{F}^{\kappa}$ the
$\mathcal{M}_{p}$-submodule of $\mathcal{M}^{n+1}_{p}$ generated by
$\\{(\mathcal{D}^{\alpha}f_{0},\ldots,\mathcal{D}^{\alpha}f_{n}):|\alpha|\leq\kappa\\}$
for $\kappa>0$ and $\mathcal{F}^{-1}=\\{0\\}$. Here
$|\alpha|=\sum_{j=1}^{m}\alpha_{j}$ and
$\mathcal{D}^{\alpha}\varphi=\frac{\partial^{|\alpha|}\varphi}{\partial
z_{1}^{\alpha_{1}}\ldots\partial z_{m}^{\alpha_{m}}}$ for each meromorphic
function $\varphi$ on $U$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{N}^{m}$. We define

$\displaystyle
r_{f}(k)=\text{rank}_{\mathcal{M}_{p}}\mathcal{F}_{p}^{k}-\text{rank}_{\mathcal{M}_{p}}\mathcal{F}_{p}^{k-1}\
(k\geq 0),$ $\displaystyle r_{f}:=\sum_{k\geq 0}r_{f}(k)-1\ \ \text{and }\
\ell_{f}:=\sum_{k\geq 0}kr_{f}(k),$ $\displaystyle
m_{f}:=\sum_{k,l}(k-l)^{+}\min\left\\{{}_{n-1}H_{l},\left(r_{f}(k)-\sum_{\lambda=0}^{l-1}{}_{n-1}H_{\lambda}\right)^{+}\right\\},$

where $x^{+}=\max\\{x,0\\}$ for a real number $x$ and ${}_{n-1}H_{\lambda}$
denotes the number of repeated combinations of $\lambda$ elements among $n-1$
elements. One has

$0\leq m_{f}\leq\ell_{f}\leq\frac{n(n+1)}{2}.$

Let $H$ be a hyperplane in ${\mathbb{P}}^{n}({\mathbb{C}})$, we (throughout
this paper) also denote by the same letter $H$ a linear form defining $H$,
i.e., we may write

$H(x_{0},\ldots,x_{n})=\sum_{j=0}^{n}a_{1j}x_{j},$

Let $f=(f_{0}:\cdots:f_{n})$ be a locally reduced representation of $f$. We
set

$H(f):=a_{0}f_{0}+\cdots+a_{n}f_{n}.$

Then, the function $H(f)$ is locally defined and depends on the choice of the
local reduced representation of $f$. However, its zero divisor $\nu_{H(f)}$
does not depend on this choice and hence it is globally well-defined.

Two meromorphic functions $f$ and $g$ from $M$ into
${\mathbb{P}}^{n}({\mathbb{C}})$ are said to share the hyperplane $H$ without
multiplicity if $f^{-1}(H)=g^{-1}(H)$ and $f=g$ on $f^{-1}(H)$. In 1986, H.
Fujimoto [5] proved the following uniqueness theorem for meromorphic mappings
on a complete Kähler manifold sharing a family of hyperplanes in
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position as follows.

Theorem A (see [5, Main Theorem]). Let $M$ be a complete, connected Kähler
manifold whose universal covering is biholomorphic to
$\mathbb{B}(R_{0})\subset{\mathbb{C}}^{m}\ (0<R_{0}\leq+\infty)$, and let $f$
and $g$ be linearly nondegenerate meromorphic maps of $M$ into
${\mathbb{P}}^{n}({\mathbb{C}})$. If $f$ and $g$ satisfy the condition
$(C_{\rho})$ and there exist hyperplanes $\\{H_{i}\\}_{i=1}^{q}$ of
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position such that

  * (i)

$f^{-1}(H_{i})=g^{-1}(H_{i})\ (1\leq i\leq q)$ and $f=g$ on
$\cup_{i=1}^{q}f^{-1}(H_{i}),$

  * (ii)

$q>n+1+\rho(\ell_{f}+\ell_{g})+m_{f}+m_{g},$

then $f\equiv g$.

Hence, Theorem A implies the previous uniqueness results for meromorphic
mappings from ${\mathbb{C}}^{m}$ into ${\mathbb{P}}^{n}({\mathbb{C}})$ given
by R. Nevanlinna [8], L. Smiley [12] and S. J. Drouilhet [2]. Recently, K.
Zhou and L. Jin [14] considered the case of meromorphic mappings from
${\mathbb{C}}^{m}$ into ${\mathbb{P}}^{n}({\mathbb{C}})$ where the condition
“$f^{-1}(H_{i})=g^{-1}(H_{i})$” is replaced by a weaker one that
$f^{H_{i}}\subset g^{-1}(H_{i})$ for some hyperplane $H_{i}$. They proved the
following.

Theorem B (see [14, Theorem 1.1]). Let
$f,g:{\mathbb{C}}^{m}\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be meromorphic
maps. Let $H_{1},\ldots,H_{q}$ be hyperplanes of
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position with
$f({\mathbb{C}}^{m})\nsubseteq H_{j},g({\mathbb{C}}^{m})\nsubseteq H_{j}$ for
$1\leq j\leq q$ and $\dim f^{-1}(H_{i}\cap H_{j})\leq m-2$ for $1\leq i<j\leq
q$. Suppose that:

  * (a)

$f^{-1}(H_{j})=g^{-1}(H_{j})$ for $1\leq j\leq p$, and $f^{-1}(H_{j})\subseteq
g^{-1}(H_{j})$ for $p<j\leq q$,

  * (b)

$f\equiv g$ on $\bigcup_{j=1}^{q}f^{-1}(H_{j})$.

Then $f=g$ if any one of the following conditions is satisfied:

  * (i)

$f$ or $g$ is nonconstant and $p=2n+2,q>3n+3-2\sqrt{n}$.

  * (ii)

$f$ or $g$ is linearly nondegenerate and $p=2n+2,q\geq 2n+3$.

  * (iii)

$f$ or $g$ is nonconstant and $p=2n+1,q\geq 4n+3$.

  * (iv)

Both $f$ and $g$ are linearly nondegenerate and $p=n+2,q\geq n^{3}+n^{2}+n+4$.

Motivated by the work of K. Zhou and L. Jin, our first aim in this paper is to
extend the above mentioned results to the case of meromorphic mappings on
Kähler manifold. Namely, we will prove the following result.

######  Theorem 1.1.

Let $M$ be a complete, connected Kähler manifold whose universal covering is
biholomorphic to $\mathbb{B}(R_{0})\subset{\mathbb{C}}^{m}\
(0<R_{0}\leq+\infty)$. Let $f,g:M\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be
linearly nondegenerate meromorphic mappings satisfying a condition
$(C_{\rho})$. Let $H_{1},\ldots,H_{q}$ be hyperplanes of
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position with $\dim
f^{-1}(H_{i}\cap H_{j})\leq m-2$ for every $1\leq i<j\leq q$, such that

  * (i)

$f^{-1}(H_{i})=g^{-1}(H_{i})\ \forall\ 1\leq i\leq p$, $f^{-1}(H_{i})\subset
g^{-1}(H_{i})\ \forall\ p+1\leq i\leq q$,

  * (ii)

$f=g$ on $\bigcup_{i=1}^{q}f^{-1}(H_{i})$,

where $n+2\leq p\leq 2n+2$. Then $f\equiv g$ if

$q>2n+2+pn\left(\frac{n+1}{p-n-1}-1\right)+2\rho\left(\ell_{f}+\frac{n+1}{p-n-1}\ell_{g}\right)$
$\text{or }\
q>2n+1+pn\left(\frac{n}{p-n-1}-\frac{n-1}{n}\right)+2\rho\left(\ell_{f}+\frac{n}{p-n-1}\ell_{g}\right).$

Remark 1. The condition of $p$ and $q$ is fullfiled in the following cases:

  * (1)

$p=2n+2$, $q>2n+2+2\rho(\ell_{f}+\ell_{g})$.

  * (2)

$p=2n+1$, $q>4n+2+2\rho(\ell_{f}+\ell_{g})$.

  * (3)

$p=2n$,
$q>6n+2+\frac{2}{n-1}+2\rho\left(\ell_{f}+\frac{n}{n-1}\ell_{g}\right)$, for
$n\geq 2$.

  * (4)

$p=n+2$, $q>n^{3}+n^{2}+n+3+2\rho\left(\ell_{f}+n\ell_{g}\right)$.

Then, our result implies the above mentioned result of K. Zhou and L. Jin for
the case of linearly nondegenerate meromorphic mappings. In order to prove the
above result, we have to develop our previous method on “funtions of small
integration” and “functions of bounded integeration” in [9, 10]. We will prove
a key lemma (see Lemma 3.1 in Section 3), which generalizes and improves
Proposition 3.5 of [10], and apply it to estimate the divisor of the
auxilliary functions.

With the useful of Lemma 3.1, we may improve the previous result on the
uniqueness problem and the algebraic degeneracy problem of meromorphic
mappings on Kähler manifolds. Moreover, we may consider the case of
meromorphic mappings satisfying the condition $(C_{\rho})$ with different
numbers $\rho$. Namely, we will prove the following result.

######  Theorem 1.2.

Let $M$ be as in Theorem 1.1. Let
$f,g:M\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be linearly nondegenerate
meromorphic mappings, which satisfy the condition $(C_{\rho_{f}})$ and
$(C_{\rho_{g}})$ for non-negative constants $\rho_{f}$ and $\rho_{g}$,
respectively. Let $H_{1},\ldots,H_{q}$ be $q$ hyperplanes of
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position with $\dim
f^{-1}(H_{i}\cap H_{j})\leq m-2$ for every $1\leq i<j\leq q$, such that

  * (i)

$\nu^{[\ell]}_{H_{i}(f)}=\nu^{[\ell]}_{H_{i}(g)}$ for every $i=1,\ldots,q,$

  * (ii)

$f=g$ on $\bigcup_{1\leq i\leq q}(f^{1})^{-1}(H_{i})$.

Then $f\equiv g$ if any one of the following conditions is satisfied:

  * (a)

$\ell=1$ and $q>2n+2+\rho_{f,g}(\ell_{f}+\ell_{g}),$ where
$\rho_{f,g}=2\frac{\rho_{f}\cdot\rho_{g}}{\rho_{f}+\rho_{g}}$.

  * (b)

$\ell\geq n+1$ and
$q>2n+1+\max\big{\\{}\frac{3(p-2)p-(\ell-n)}{3(p-2)p+(\ell-n)/n},\frac{2n-1}{2n-1/n}\big{\\}}$
$+\rho_{f,g}(\frac{4n^{2}-1}{2n^{2}-1}(\ell_{f}+\ell_{g})+\ell-n),$ where
$p=\binom{n+1}{2n+2}$.

For the case of more than two meromorphic mappings sharing a family of
hyperplanes, we prove the following two results.

######  Theorem 1.3.

Let $M$ be as in Theorem 1.1. Let
$f^{1},f^{2},f^{3}:M\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})\ (n\geq 2)$ be
linearly nondegenerate meromorphic mappings, which satisfy the condition
$(C_{\rho_{f^{1}}}),(C_{\rho_{f^{2}}})$ and $(C_{\rho_{f^{3}}})$ for non-
negative constants $\rho_{f^{1}},\rho_{f^{2}}$ and $\rho_{f^{3}}$,
respectively. Let $H_{1},\ldots,H_{q}$ be $q$ hyperplanes of
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position with
$\dim(f^{1})^{-1}(H_{i}\cap H_{j})\leq m-2$ for every $1\leq i<j\leq q$.
Assume that $f=g$ on $\bigcup_{{\mathrel{\mathop{{1\leq i\leq
q}}\limits^{1\leq u\leq 3}}}}(f^{u})^{-1}(H_{i})$. Then $f^{1}\wedge
f^{2}\wedge f^{3}=0$ if
$q>n+1+\frac{3nq}{2q+2n-2}+2\left(\sum_{u=1}^{3}\frac{1}{\rho_{f^{u}}}\right)^{-1}\sum_{u=1}^{3}\ell_{f^{u}}.$

######  Theorem 1.4.

Let $M$ be as in Theorem 1.1. Let
$f^{1},\ldots,f^{k}:M\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be $k$
linearly nondegenerate meromorphic mappings, which satisfy the conditions
$(C_{\rho_{f^{u}}}),\ldots,(C_{\rho_{f^{k}}})$ for non-negative constants
$\rho_{f^{1}},\ldots,\rho_{f^{k}}$, respectively. Let $H_{1},\ldots,H_{q}$ be
$q$ hyperplanes of ${\mathbb{P}}^{n}({\mathbb{C}})$ in general position with
$\dim(f^{1})^{-1}(H_{i}\cap H_{j})\leq m-2$ for every $1\leq i<j\leq q$ such
that

  * (i)

$\nu^{[n]}_{H_{i}(f^{1})}=\cdots=\nu^{[n]}_{H_{i}(f^{u})}$ for every
$i=1,\ldots,q,$

  * (ii)

$f=g$ on $\bigcup_{1\leq i\leq q}(f^{1})^{-1}(H_{i})$.

Then $f^{1}\wedge\cdots\wedge f^{k}=0$ if
$q>n+1+\frac{knq}{(k-1)q+k(n-1)}+2\left(\sum_{u=1}^{k}\frac{1}{\rho_{f^{u}}}\right)^{-1}\sum_{u=1}^{k}\ell_{f^{u}}.$

Remark 2. (a) Theorems 1.2, 1.3 and 1.4 generalize and improve the recent
results in [10] (Lemma 4.3, Theorem 1.2) and [11] (Theorem 1.2).

(b) In this paper, we only concentrate on linearly nondegenerate meromorphic
mappings. But our method can be applied to study the case of nonconstant
meromorphic mappings. However, in that case the computation may be much more
complicate, since there are many more parameters appearing. Then, that problem
is still an interesting open question.

##  2\. Auxiliary results

Let $E$ be a divisor on $\mathbb{B}(R_{0})$, which is usually regarded as a
function from $\mathbb{B}(R_{0})$ into $\mathbb{Z}$. The support
$\operatorname{Supp}(E)$ is defined as the closure of the set $\\{z|E(z)\neq
0\\}$. For a positive integer $k$ (may be $+\infty$), we define
$E^{[k]}(z)=\min\\{E(z),k\\}$ and

$n^{[k]}(t,E):=\begin{cases}\int_{\operatorname{Supp}(E)\cap
B(t)}E^{[k]}v_{m-1}&\text{ if }m\geq 2,\text{ where
}v_{m-1}(z)=\left(dd^{c}\|z\|^{2}\right)^{m-1},\\\ \sum_{|z|\leq
t}E^{[k]}(z)&\text{ if }m=1.\end{cases}$

The truncated counting function to level $k$ of $E$ is defined by

$N^{[k]}\left(r,r_{0};E\right):=\int_{r_{0}}^{r}\frac{n^{[k]}(t,E)}{t^{2m-1}}dt\quad\left(r_{0}<r<R_{0}\right).$

We omit the character [k] if $k=+\infty$.

Let $\varphi$ be a non-zero meromorphic function on $\mathbb{B}(R_{0})$. We
denote by $\nu_{\varphi}^{0}$ (resp. $\nu_{\varphi}^{\infty}$ ) the divisor of
zeros (resp. divisor of poles ) of $\varphi$ and set
$\nu_{\varphi}=\nu_{\varphi}^{0}-\nu_{\varphi}^{\infty}$. For convenience, we
will write $N_{\varphi}\left(r,r_{0}\right)$ and
$N_{\varphi}^{[k]}\left(r,r_{0}\right)$ for
$N\left(r,r_{0};\nu_{\varphi}^{0}\right)$ and
$N^{[k]}\left(r,r_{0};\nu_{\varphi}^{0}\right)$, respectively.

Let $f:\mathbb{B}(R_{0})\longrightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a
meromorphic mapping. Fix a homogeneous coordinates system
$\left(w_{0}:\cdots:w_{n}\right)$ on ${\mathbb{P}}^{n}({\mathbb{C}})$. We take
a reduced representation $f=\left(f_{0}:\cdots:f_{n}\right)$ and set
$\|f\|=\left(\left|f_{0}\right|^{2}+\cdots+\left|f_{n}\right|^{2}\right)^{1/2}$.
The characteristic function of $f$ is defined by

$T_{f}(r,r_{0})=\int_{\|z\|=r}\log\|f\|\sigma_{m}-\int_{\|z\|=r_{0}}\log\|f\|\sigma_{m},\quad
0<r_{0}<r<R_{0},$

where $\sigma_{m}(z)=d^{c}\log\|z\|^{2}\wedge(dd^{c}\log\|z\|^{2})^{m-1}.$
Here and throughout this paper, we assume that the numbers $r_{0}$ and $R_{0}$
are fixed with $0<r_{0}<R_{0}$.

Suppose that $f$ is linearly nondegenerate. Then, there is an $n+1$-tuple
$\alpha=(\alpha_{0},\ldots,\alpha_{n})\in(\mathbb{N}^{m})^{n+1}$ such that
$(\mathcal{D}^{\alpha_{0}}(f),\ldots,\mathcal{D}^{\alpha_{\ell(k)-1}})$ is a
basis of $\mathcal{M}_{p}$-module $\mathcal{F}^{\ell(k)}$, where
$\ell(k)=\dim_{\mathcal{M}_{p}}\mathcal{F}^{k}$ for all
$k=1,\ldots,k_{0}=\min\\{k:\ell(k)=n+1\\}$. The tuple
$\alpha=(\alpha_{0},\ldots,\alpha_{n})$ is called the admissible set of $f$
and

$W_{\alpha_{0},\ldots,\alpha_{n}}(f_{0},\ldots,f_{n}):=\det(\mathcal{D}^{\alpha_{j}}(f_{i}))_{0\leq
j,i\leq n}$

is called the generalized Wronskian of $f$. We note that
$|\alpha|=\sum_{i=0}^{n}|\alpha_{i}|=\ell_{f}.$

######  Proposition 2.1 (see [5, Proposition 2.12]).

Let $H_{1},\ldots,H_{q}$ be $q$ hyperplanes in
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position. Let $f$ be a linearly
nondegenerate meromorphic mapping from the ball
$\mathbb{B}^{m}(R_{0})\subset{\mathbb{C}}^{m}$ into
${\mathbb{P}}^{n}({\mathbb{C}})$ with a reduced representation
$f=(f_{0},\ldots,f_{n})$ and let $(\alpha_{0},\ldots,\alpha_{n})$ be an
admissible set of $f$. Then, for $0<r_{0}<R_{0}$ and $0<t\ell_{f}<p<1$, there
exists a positive constant $K$ such that for $r_{0}<r<R<R_{0}$,

$\int_{S(r)}\biggl{|}z^{\alpha_{0}+\cdots+\alpha_{n}}\dfrac{W_{\alpha_{0},\ldots,\alpha_{n}}(f_{0},\ldots,f_{n})}{H_{1}(f)\ldots
H_{q}(f)}\biggl{|}^{t}\cdot\|f\|^{t(q-n-1)}\sigma_{m}\leq
K\biggl{(}\dfrac{R^{2m-1}}{R-r}T_{f}(R,r_{0})\biggl{)}^{p}.$

##  3\. Main lemmas

Let $f^{1},\ldots,f^{k}$ be $k$ meromorphic mappings from
$\mathbb{B}^{m}(R_{0})$ into ${\mathbb{P}}^{n}({\mathbb{C}})$. We fix a
reduced representation $f^{u}=(f^{u}_{0}:\cdots:f^{u}_{n})$ of $f^{u}$ and set
$\|f^{u}\|=(|f^{u}_{0}|^{2}+\cdots+|f^{u}_{n}|^{2})^{1/2}$ for $u=1,\ldots,k$.

Denote by $\mathcal{C}(\mathbb{B}^{m}(R_{0}))$ the set of all functions
$g:\mathbb{B}^{m}(R_{0})\to[0,+\infty]$ which is continuous outside an
analytic set of codimension two and only attain $+\infty$ in an analytic thin
set.

For a non negative integer $l_{0}$, we denote by $S(l_{0};f^{1},\ldots,f^{k})$
the set of all functions $g$ in $\mathcal{C}(\mathbb{B}^{m}(R_{0}))$ such that
there exist an element
$\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbb{N}^{m}$ with $|\alpha|\leq
l_{0}$ satisfying: for every $0\leq tl_{0}<p<1$, there exists a positive
number $K$ with

$\int_{S(r)}|z^{\alpha}g|^{t}\sigma_{m}\leq
K\left(\frac{R^{2m-1}}{R-r}\sum_{u=1}^{k}T_{f^{u}}(r,r_{0})\right)^{p}$

for all $r$ with $0<r_{0}<r<R<R_{0}$, where
$z^{\alpha}=z_{1}^{\alpha_{1}}\cdots z_{m}^{\alpha_{m}}$.

Let $p$ be a non negative integer. Denote by
$B(l_{0};(f^{1},p_{1}),\ldots,(f^{k},p_{k}))$ the set of all meromorphic
functions $h$ on $\mathbb{B}^{m}(R_{0})$ such that there exists $g\in
S(l_{0};f^{1},\ldots,f^{k})$ satisfying

$|h|\leq\|f^{1}\|^{p_{1}}\cdots\|f^{k}\|^{p_{k}}\cdot g,$

outside a proper analytic subset of $\mathbb{B}^{m}(R_{0})$. We will write
$B(p,l_{0};f^{1},\ldots,f^{k})$ as in [10] for
$B(l_{0};(f^{1},p),\ldots,(f^{k},p))$.

We easily have the following fundamental properties of the families
$S(l_{0};\\{f^{u}\\}_{u=1}^{k})$ and $B(l_{0};\\{(f^{u},p_{u})\\}_{u=1}^{k})$.

  * •

If $g$ is a constant function then $g\in S(0;\\{f^{u}\\}_{u=1}^{k})$.

  * •

If $g_{i}\in S(l_{i};\\{f^{u}\\}_{u=1}^{k})\ (1\leq i\leq s)$ then
$\prod_{i=1}^{s}g_{i}\in S(\sum_{i=1}^{s}l_{i};\\{f^{u}\\}_{u=1}^{k})$ (see
Proposition 2.1 in [10]).

  * •

A meromorphic mapping $h$ belongs to $B(0,l_{0};\\{f^{u}\\}_{u=1}^{k})$ if and
only if $|h|\in S(l_{0};\\{f^{u}\\}_{u=1}^{k})$.

  * •

If $h_{i}\in B(l_{i};\\{f^{1},p^{i}_{1}\\},\ldots,\\{f^{k},p^{i}_{k}\\})\
(1\leq i\leq s)$ then

$h_{1}\cdots h_{m}\in
B(\sum_{i=1}^{s}l_{i};\\{f^{1},\sum_{i=1}^{s}p^{i}_{1}\\},\ldots,\\{f^{k},\sum_{i=1}^{s}p^{i}_{k}\\}).$

  * •

Proposition 2.1 implies that if $W(f)$ is the generalized wronskian of a
linearly nondegenerate meromorphic mapping
$f:\mathbb{B}^{m}(R_{0})\to{\mathbb{P}}^{n}({\mathbb{C}})$ and
$H_{1},\ldots,H_{q}$ be $q$ hyperplanes in ${\mathbb{P}}^{n}({\mathbb{C}})$ in
general position then the function
$\left|\dfrac{W(f)\cdot\|f\|^{q-n-1}}{H_{1}(f)\ldots H_{q}(f)}\right|$ belongs
to $S(\ell_{f};f)$.

Firstly, we prove the following key lemma.

######  Lemma 3.1.

Let $M=\mathbb{B}^{m}(R_{0})\ (0<R_{0}\leq+\infty)$ be a complete connected
Kähler manifold. Let $k$ be a positive integer and for each
$u\in\\{1,\ldots,k\\}$, let $f^{u}$ be a linearly nondegenerate meromorphic
mapping from $M$ into ${\mathbb{P}}^{n}({\mathbb{C}})$, which satisfies the
condition $(C_{\rho_{u}})$ and has a reduced representation
$f^{u}=(f_{0}^{u}:\cdots:f_{n}^{u})$. Let
$\\{H^{u}_{1},\ldots,H^{u}_{q_{u}}\\}\ (1\leq u\leq k)$ be $k$ families of
hyperplanes of ${\mathbb{P}}^{n}({\mathbb{C}})$ in general position, where
$q_{1},\ldots,q_{k}$ are positive integers. Assume that there exists a non
zero holomorphic function $h$ on $\mathbb{B}({\mathbb{R}}_{0})$ such that:

  * (a)

$h\in B(l_{0};\\{(f^{u},p_{u})\\}_{u=1}^{k}),$ where $l_{0}$ is a non-negative
integer, $p_{1},\ldots,p_{k}$ are positive constants;

  * (b)

$\nu_{h}\geq\sum_{u=1}^{k}\lambda_{u}\sum_{i=1}^{q_{u}}\nu^{[n]}_{H^{u}_{i}(f^{u})},$
where $\lambda_{u}(1\leq u\leq k)$ are positive constants.

Then there is an index $u$ such that $\lambda_{u}(q_{u}-n-1)-p_{u}\leq 0,$ or

$\sum_{u=1}^{k}\left[\frac{\lambda_{u}(q_{u}-n-1)-p_{u}}{\rho_{u}}-2\lambda_{u}\ell_{f^{u}}\right]\leq
2l_{0}.$

Here, we regard that $\frac{x}{0}=+\infty$ and $\frac{-x}{0}=-\infty$ for
every $x>0$.

###### Proof.

Suppose contrarily that $\lambda_{u}(q_{u}-n-1)-p_{u}>0,$ for all
$u=1,\ldots,k$ and

$\sum_{u=1}^{k}\left[\frac{\lambda_{u}(q_{u}-n-1)-p_{u}}{\rho_{u}}-2\lambda_{u}\ell_{f^{u}}\right]>2l_{0}.$

We consider the following two cases.

Case 1: $R_{0}=+\infty$. By the second main theorem in Nevanlinna theory we
have

$\displaystyle\sum_{u=1}^{k}\lambda_{u}(q_{u}-n-1)T_{f^{u}}(r,1)$
$\displaystyle\leq\sum_{u=1}^{k}\lambda_{u}\sum_{i=1}^{q_{u}}N^{[n]}_{H^{u}_{i}(f^{u})}(r,1)+o(\sum_{u=1}^{k}T_{f^{u}}(r,1))$
$\displaystyle\leq N_{h}(r,1)+o(\sum_{u=1}^{k}T_{f^{u}}(r,1))$
$\displaystyle=\sum_{u=1}^{k}p_{u}T_{f^{u}}(r,1)+o(\sum_{u=1}^{k}T_{f^{u}}(r,1)),$

for all $r\in(1,+\infty)$ outside a Lebesgue set of finite measure. This is a
contradiction.

Case 2: $R_{0}<+\infty$. For each $u\ (1\leq u\leq k)$, since $f^{u}$ is
linearly nondegenerate, there exists an admissible set
$(\alpha_{0}^{u},\ldots,\alpha_{n}^{u})\in\mathbb{(}\mathbb{N}^{m})^{n+1}$
such that the generalized Wronskian

$W(f^{u}):=\det\left(\mathcal{D}^{\alpha_{i}^{u}}(f_{j}^{u});0\leq i,j\leq
n\right)\not\equiv 0.$

By usual argument in Nevanlinna theory, we have

$\nu_{h}\geq\sum_{u=1}^{k}\lambda_{u}\sum_{i=1}^{q}\nu^{[n]}_{H^{u}_{i}(f^{u})}\geq\sum_{u=1}^{k}\lambda_{u}\left(\sum_{i=1}^{q}\nu_{H^{u}_{i}(f^{u})}-\nu_{W(f^{u})}\right).$

Put
$w_{u}(z):=z^{\alpha_{0}^{u}+\cdots+\alpha_{n}^{u}}\dfrac{W(f^{u})}{\prod_{i=1}^{q}H^{u}_{i}(f^{u})}\
(1\leq u\leq k)$.

Since $h\in B(l_{0};\\{(f^{u},p_{u})\\}_{u=1}^{k})$, there exists a non-
negative plurisubharmonic function $g\in S(l_{0};f^{1},\ldots,f^{k})$ and
$\beta=(\beta_{1},\ldots,\beta_{m})\in\mathbb{Z}^{m}_{+}$ with $|\beta|\leq
l_{0}$ such that

(3.2)
$\displaystyle\int_{S(r)}\left|z^{\beta}g\right|^{t^{\prime}}\sigma_{m}=O\left(\frac{R^{2m-1}}{R-r}\sum_{u=1}^{k}T_{f^{u}}(r,r_{0})\right)^{l},$

for every $0\leq l_{0}t^{\prime}<l<1$ and
$|h|\leq\prod_{u=1}^{k}\|f^{u}\|^{p_{k}}g.$ Put
$t=\frac{2}{\sum_{u=1}^{k}(\lambda_{u}(q_{u}-n-1)-p_{u})/\rho_{u}}>0$ and
$\phi:=|w_{1}|^{\lambda_{1}}\cdots|w_{k}|^{\lambda_{k}}\cdot|z^{\beta}h|,$
Then $a=t\log\phi$ is a plurisubharmonic function on $\mathbb{B}^{m}(R_{0})$
and

$(\sum_{u=1}^{k}\lambda_{u}\ell_{f^{u}}+l_{0})t<1.$

Therefore, we may choose a positive number $p$ such that
$0<\sum_{u=1}^{k}\lambda_{u}t<p<1.$

Since $f^{u}$ satisfies the condition $(C_{\rho_{u}})$, there is a continuous
plurisubharmonic function $\varphi_{u}$ on $\mathbb{B}^{m}(R_{0})$ such that

$e^{\varphi_{u}}dV\leq\|f^{u}\|^{2\rho_{u}}v_{m}.$

Note that in this case, $\rho_{u}>0$ for all $u$. Then the function
$\varphi=\lambda^{\prime}_{1}\varphi_{1}+\cdots+\lambda^{\prime}_{k}\varphi_{k}+a$
is a plurisubharmonic on $\mathbb{B}^{m}(R_{0})$, where
$\lambda_{u}^{\prime}=\frac{(\lambda_{u}(q-n-1)-p_{u})t}{2\rho_{u}}$. One has
$\sum_{u=1}^{k}\lambda_{u}^{\prime}=1$ and then

$\displaystyle
e^{\varphi}dV=e^{\lambda^{\prime}_{1}\varphi_{1}+\cdots+\lambda^{\prime}_{k}\varphi_{k}+t\log\phi}dV\leq
C^{\prime}\cdot
e^{t\log\phi}\cdot\prod_{u=1}^{k}\|f^{u}\|^{2\lambda^{\prime}_{u}\rho_{u}}v_{m}=C^{\prime}\cdot|\phi|^{t}\cdot\prod_{u=1}^{k}\|f^{u}\|^{2\lambda^{\prime}_{u}\rho_{u}}v_{m}$
$\displaystyle=C^{\prime\prime}\cdot|z^{\beta}g|^{t}\cdot\prod_{u=1}^{k}(|w_{u}|^{\lambda_{u}t}\|f^{u}\|^{2\lambda_{u}^{\prime}\rho_{u}+p_{u}t})v_{m}=C^{\prime\prime}\cdot|z^{\beta}g|^{t}\cdot\prod_{u=1}^{k}(|w_{u}|\cdot\|f^{u}\|^{(q_{u}-n-1)})^{t\lambda_{u}}v_{m},$

where $C^{\prime},C^{\prime\prime}$ are positive constants. Setting
$x_{u}=\frac{\sum_{i=1}^{k}\lambda_{i}\ell_{f^{i}}+l_{0}}{\lambda_{u}\ell_{f^{u}}}$,
$y=\frac{\sum_{i=1}^{k}\lambda_{i}\ell_{f^{i}}+l_{0}}{l_{0}}$, we have
$\sum_{u=1}^{k}\frac{1}{x_{u}}+\frac{1}{l_{0}}=1$. By integrating both sides
of the above inequality over $\mathbb{B}^{m}(R_{0})$ and applying Hölder
inequality, we get

$\displaystyle\int_{\mathbb{B}^{m}(R_{0})}e^{\varphi}dV$ $\displaystyle\leq
C^{\prime\prime}\left(\int_{\mathbb{B}^{m}(R_{0})}|z^{\beta}g|^{ty}\right)^{1/y}\cdot\prod_{u=1}^{k}\left(\int_{\mathbb{B}^{m}(R_{0})}(|w_{u}|\cdot\|f^{u}\|^{(q_{u}-n-1)})^{\lambda_{u}tx_{u}}v_{m}\right)^{1/x_{u}}$
(3.3)
$\displaystyle\begin{split}&=C^{\prime\prime}\left(2m\int_{0}^{1}r^{2m-1}\int_{S(r)}|z^{\beta}g|^{ty}v_{m}\right)^{1/y}\\\
&\times\prod_{u=1}^{k}\left(2m\int_{0}^{1}r^{2m-1}\int_{S(r)}(|w_{u}|\cdot\|f^{u}\|^{(q_{u}-n-1)})^{\lambda_{u}x_{u}t}v_{m}\right)^{1/x_{u}}.\end{split}$

Subcase 2.a: We suppose that

$\lim\limits_{r\rightarrow
R_{0}}\sup\dfrac{\sum_{u=1}^{k}T_{f^{u}}(r,r_{0})}{\log 1/(R_{0}-r)}<\infty.$

We see that
$\lambda_{u}tx_{u}\ell_{f^{u}}=tyl_{0}=t(\sum_{i=1}^{k}\lambda_{i}\ell_{f^{i}}+l_{0})<p$.
By Proposition 2.1, there exists a positive constant $K$ such that, for every
$0<r_{0}<r<R<R_{0},$ we have

$\displaystyle\int_{S(r)}\bigl{(}|w_{u}|\cdot\|f^{u}\|^{(q_{u}-n-1)}\bigl{)}^{\lambda_{u}x_{u}t}\sigma_{m}\leq
K\left(\dfrac{R^{2m-1}}{R-r}T_{f^{u}}(R,r_{0})\right)^{p}\ (1\leq u\leq k),$
and $\displaystyle\int_{S(r)}|z^{\beta}g|^{ty}v_{m}\leq
K\left(\dfrac{R^{2m-1}}{R-r}T_{f^{u}}(R,r_{0})\right)^{p}.$

Choosing $R=r+\dfrac{R_{0}-r}{e\max_{1\leq u\leq k}T_{f^{u}}(r,r_{0})}$, we
have $T_{f^{u}}(R,r_{0})\leq 2T_{f^{u}}(r,r_{0})$, for all $r$ outside a
subset $E$ of $(0,R_{0}]$ with $\int_{E}\frac{1}{R_{0}-r}dr<+\infty$. Then,
the above inequality implies that

$\displaystyle\int_{S(r)}\bigl{(}|w_{u}|\cdot\|f^{u}\|^{(q_{u}-n-1)}\bigl{)}^{\lambda_{u}x_{u}t}\sigma_{m}\leq\dfrac{K^{\prime}}{(R_{0}-r)^{p}}\left(\log\frac{1}{R_{0}-r}\right)^{2p}\
(1\leq u\leq k),$ and
$\displaystyle\int_{S(r)}|z^{\beta}g|^{ty}v_{m}\leq\dfrac{K^{\prime}}{(R_{0}-r)^{p}}\left(\log\frac{1}{R_{0}-r}\right)^{2p}$

for all $r$ outside $E$, and for some positive constant $K^{\prime}$. The
inequality (3.3) yields that

$\displaystyle\int_{\mathbb{B}^{m}(R_{0})}e^{u}dV$ $\displaystyle\leq
C^{\prime\prime}2m\int_{0}^{R_{0}}r^{2m-1}\frac{K^{\prime}}{R_{0}-r}\left(\log\frac{1}{R_{0}-r}\right)^{2p}dr<+\infty.$

This contradicts the results of S.T. Yau [13] and L. Karp [6].

Subcase 2.b: We suppose that

$\lim\limits_{r\rightarrow
R_{0}}\sup\dfrac{\sum_{u=1}^{k}T_{f^{u}}(r,r_{0})}{\log 1/(R_{0}-r)}=\infty.$

By [4, Proposition 6.2], we have

$\displaystyle\sum_{u=1}^{k}p_{u}T_{f^{u}}(r,r_{0})$ $\displaystyle\geq
N_{h}(r,r_{0})+S(r)\geq\sum_{u=1}^{k}\lambda_{p}\sum_{i=1}^{q}N_{H^{u}_{i}(f^{u})}^{[n]}(r,r_{0})+S(r)$
$\displaystyle\geq\sum_{u=1}^{k}\lambda_{u}(q_{u}-n-1)T_{f^{u}}(r,r_{0})+O\bigl{(}\log^{+}\frac{1}{R_{0}-r}+\log^{+}\sum_{u=1}^{k}T_{f^{u}}(r_{0},r)\bigl{)},$

for every $r$ excluding a set $E$ with $\int_{E}\frac{dr}{R_{0}-r}<+\infty$.
This is a contradiction.

Hence, the supposition is false. The lemma is proved. ∎

Secondly, we prove the following generalization theorem for uniqueness problem
of linearly nondegenerate meromorphic mappings on Kähler manifolds.

######  Lemma 3.4.

Let $M$ be a complete, connected Kähler manifold whose universal covering is
biholomorphic to $\mathbb{B}(R_{0})\subset{\mathbb{C}}^{m}$, where
$0<R_{0}\leq\infty$. Let $f,g:M\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be
linearly nondegenerate meromorphic mappings, which satisfy the condition
$(C_{\rho_{f}}),(C_{\rho_{g}})$ for non-negative constants
$\rho_{f},\rho_{g}$, respectively. Let $H_{1},\ldots,H_{q}$ be $q$ hyperplane
of ${\mathbb{P}}^{n}({\mathbb{C}})$ in general position and let $n+2\leq p\leq
2n+2<q$. Assume that:

  * (a)

$\dim f^{-1}(H_{i}\cap H_{j})\leq m-2\ \forall\ 1\leq i<j\leq q,$

  * (b)

$f^{-1}(H_{i})=g^{-1}(H_{i})\ \forall\ 1\leq i\leq p,\ f^{-1}(H_{i})\subset
g^{-1}(H_{i})\ \forall\ p+1\leq i\leq q,$

  * (c)

$f=g$ on $\bigcup_{i=1}^{q}f^{-1}(H_{i}).$

Then $f\equiv g$ if there exist non-negative numbers $t\leq\frac{2}{n}$ and
$\alpha$ such that:

  * (1)

$(2+t)(q-n-1)-(2n+2+p(t+\alpha))>0,$

  * (2)

$\left(2+\frac{\alpha}{n}\right)(p-n-1)-(2n+2)>0,$

  * (3)

$\frac{(2+t)(q-n-1)-(2n+2+p(t+\alpha))}{\rho_{f}}+\frac{\left(2+\frac{\alpha}{n}\right)(p-n-1)-(2n+2)}{\rho_{g}}>2\left((2+t)\ell_{f}+(2+\frac{\alpha}{n})\ell_{g}\right),$

where $t=0$ if $p>q-n-1$ and $t=\frac{2}{n}$ if $p\leq q-n-1$.

###### Proof.

Since the universal covering of $M$ is biholomorphic to
$\mathbb{B}(R_{0}),0<R_{0}\leq+\infty$, by using the universal covering if
necessary, without loss of generality we assume that
$M=B(R_{0})\subset{\mathbb{C}}^{m}$. Also, we may assume that $t$ and $\alpha$
are rational numbers.

Suppose contrarily that $f\not\equiv g$. Consider the simple graph
$\mathcal{H}$, where the set of vertice is $\\{1,2,\ldots,2n+2\\}$ and the set
of edges is consist of all pairs $\\{i,j\\}$ such that
$\frac{H_{i}(f)}{H_{j}(f)}\not\equiv\frac{H_{i}(g)}{H_{j}(g)}$. Since
$f\not\equiv g$, the degree of $\mathcal{H}$ at every vertex is at least
$(2n+2)-n>\frac{2n+2}{2}$. By Dirac’s theorem, $\mathcal{H}$ has a Hamiltonian
cycle, for instance it is $(1,2,3,\ldots,2n+2,1)$. Therefore,

$P_{i}=H_{i}(f)H_{\sigma(i)}(g)-H_{i}(g)H_{\sigma(i)}(f)\not\equiv 0,$

where $\sigma(i)=i+1$ for $i<q$ and $\sigma(2n+2)=1$. We easily see that

$\nu_{P_{i}}(z)\geq\sum_{j=i,\sigma(i)}\min\left\\{\nu_{H_{j}(f)}(z),\nu_{H_{j}(g)}(z)\right\\}+\sum_{{\mathrel{\mathop{{j=1}}\limits_{{j\neq
i,\sigma(i)}}}}}^{q}\min\left\\{\nu_{H_{j}(f)}(z),1\right\\}$

for all $z$ ouside the analytic subset $\bigcup_{1\leq u<v\leq
q}f^{-1}(H_{u}\cap H_{v})$, which is of codimension two.

Then, by setting $P=\prod_{i=1}^{2n+2}P_{i}\not\equiv 0$, we have

$\displaystyle\nu_{P}$ $\displaystyle\geq
2\sum_{j=1}^{2n+2}\min\left\\{\nu_{H_{j}(f)},\nu_{H_{j}(g)}\right\\}+2n\sum_{j=1}^{2n+2}\nu^{[1]}_{H_{j}(f)}+(2n+2)\sum_{j=2n+3}^{q}\nu^{[1]}_{H_{j}(f)}$
$\displaystyle\geq
2\sum_{j=1}^{p}\left(\nu^{[n]}_{H_{j}(f)}+\nu^{[n]}_{H_{j}(g)}-n\nu^{[1]}_{H_{j}(f)}\right)+2n\sum_{j=1}^{p}\nu^{[1]}_{H_{j}(f)}+(2n+2)\sum_{j=p+1}^{q}\nu^{[1]}_{H_{j}(f)}$
$\displaystyle=2\sum_{j=1}^{p}\nu^{[n]}_{H_{j}(f)}+2\sum_{j=1}^{p}\nu^{[n]}_{H_{j}(g)}+(2n+2)\sum_{j=p+1}^{q}\nu^{[1]}_{H_{j}(f)}$
$\displaystyle\geq
2\sum_{j=1}^{q}\nu^{[n]}_{H_{j}(f)}+2\sum_{j=1}^{p}\nu^{[n]}_{H_{j}(g)}+2\sum_{j=p+1}^{q}\nu^{[1]}_{H_{j}(f)}$
$\displaystyle\geq(2+t)\sum_{j=1}^{q}\nu^{[n]}_{H_{j}(f)}-(t+\alpha)\sum_{j=1}^{p}\nu^{[n]}_{H_{j}(f)}+\alpha\sum_{j=1}^{p}\nu^{[1]}_{H_{j}(f)}+2\sum_{j=1}^{p}\nu^{[n]}_{H_{j}(g)}$
$\displaystyle\geq(2+t)\sum_{j=1}^{q}\nu^{[n]}_{H_{j}(f)}-(t+\alpha)\sum_{j=1}^{p}\nu_{H_{j}(f)}+\left(2+\frac{\alpha}{n}\right)\sum_{j=1}^{p}\nu^{[n]}_{H_{j}(g)},$

Take a positive integer $k$ so that $k(t+\alpha)$ is an integer and consider
the holomorphic function
$\tilde{P}=P^{k}\cdot\prod_{j=1}^{p}H_{j}(f)^{k(t+\alpha)}.$ It is clear that

$\nu_{\tilde{P}}\geq k(t+\alpha)\sum_{j=1}^{p}\nu_{H_{j}(f)}+k\nu_{P}\geq
k(2+t)\sum_{j=1}^{q}\nu^{[n]}_{H_{j}(f)}+k\left(2+\frac{\alpha}{n}\right)\sum_{j=1}^{p}\nu^{[n]}_{H_{j}(g)}$

and $\tilde{P}\in B\big{(}0;(f,(2n+2+p(t+\alpha))k),(g,(2n+2)k)\big{)}$. By
Lemma 3.1, one of the following must hold:

  * •

$(2+t)(q-n-1)-(2n+2+p(t+\alpha))\leq 0,$

  * •

$\left(2+\frac{\alpha}{n}\right)(p-n-1)-(2n+2)\leq 0,$

  * •

$\frac{(2+t)(q-n-1)-(2n+2+p(t+\alpha))}{\rho_{f}}+\frac{\left(2+\frac{\alpha}{n}\right)(p-n-1)-(2n+2)}{\rho_{g}}\leq
2\left((2+t)l_{f}+(2+\frac{\alpha}{n})l_{g}\right)$.

This is a contradiction. Therefore, we must have $f\equiv g$. The lemma is
proved. ∎

##  4\. Uniqueness problem

###### Proof of Theorem 1.1.

By Theorem 3.4, in order to show that $f\equiv g$ we just show the existence
of the non-negative numbers $t\leq\frac{2}{n}$ and $\alpha$ satisfying the
inequalities (1), (2), (3) of Theorem 3.4. We choose

$\alpha:=n\left(\frac{2n+2+\epsilon}{p-n-1}-2\right),$

where $\epsilon$ is a positive constant small enough. Then the inequality (2)
of Theorem 3.4 is satisfied. The inequalities (1) and (3) of Theorem 3.4
become

(4.1)
$\displaystyle(2+t)(q-n-1)-2n-2-p\left(t-2n+n\frac{2n+2+\epsilon}{p-n-1}\right)>0,$

and

(4.2)
$\displaystyle\begin{split}(2+t)(q-n-1)-2n-2-p\left(t-2n+n\frac{2n+2+\epsilon}{p-n-1}\right)+\epsilon\\\
>2\rho\left((2+t)\ell_{f}+\frac{2n+2+\epsilon}{p-n-1}\ell_{g}\right).\end{split}$

Therefore, in order to show the existence of such $t$ and $\alpha$ (equivalent
to the existence of $\epsilon>0$) we just to show that there is
$t\in[0,\frac{2}{n}]$ such that:

(4.3)
$\displaystyle\begin{split}(2+t)(q-n-1)-2n-2-&p\left(t-2n+n\frac{2n+2}{p-n-1}\right)\\\
&>2\rho\left((2+t)\ell_{f}+\frac{2n+2}{p-n-1}\ell_{g}\right).\end{split}$

If put $t=0$ then the inequality (4.3) is equivalent to that

$\displaystyle q-n-1$
$\displaystyle>n+1+pn\left(\frac{n+1}{p-n-1}-1\right)+2\rho\left(\ell_{f}+\frac{n+1}{p-n-1}\ell_{g}\right)$
$\displaystyle\Leftrightarrow\ q$
$\displaystyle>2n+2+pn\left(\frac{n+1}{p-n-1}-1\right)+2\rho\left(\ell_{f}+\frac{n+1}{p-n-1}\ell_{g}\right).$

If put $t=\frac{2}{n}$ then the inequality (4.3) is equivalent to that

$\displaystyle\frac{n+1}{n}(q-n-1)$
$\displaystyle>n+1+p\left(\frac{1-n^{2}}{n}+n\frac{n+1}{p-n-1}\right)+2\rho\left(\frac{n+1}{n}\ell_{f}+\frac{n+1}{p-n-1}\ell_{g}\right)$
$\displaystyle\Leftrightarrow\ q$
$\displaystyle>2n+1+pn\left(\frac{n}{p-n-1}-\frac{n-1}{n}\right)+2\rho\left(\ell_{f}+\frac{n}{p-n-1}\ell_{g}\right).$

Then, with the asumption of the themrem, the inequality (4.3) is satisfied for
$t=0$ or $t=\frac{2}{n}$. Hence, $f\equiv g$. The proof of the theorem is
completed. ∎

In order to prove Theorem 1.2, we need the following proposition of H.
Fujimoto [3].

######  Proposition 4.4 (See H. Fujimoto [3]).

Let $G$ be a torsion free abelian group and $A=(a_{1},\ldots,a_{q})$ be a
$q-$tuple of elements $a_{i}$ in $G$. If $A$ has the property $(P_{r,s})$ for
some $r,s$ with $q\geq r>s>1,$ then there exist $i_{1},\ldots,i_{q-r+2}$ with
$1\leq i_{1}<\cdots<i_{q-r+2}\leq q$ such that
$a_{i_{1}}=a_{i_{2}}=\cdots=a_{i_{q-r+2}}.$

Here, the $q$-tuple $A$ has the property $(P_{r,s})$ if any $r$ elements
$a_{l(1)},\ldots,a_{l(r)}$ in $A$ satisfy the condition that for any given
$i_{1},\ldots,i_{s}\ (1\leq i_{1}<\cdots<i_{s}\leq r)$, there exist
$j_{1},\ldots,j_{s}\ (1\leq j_{1}<\cdots<j_{s}\leq r)$ with
$\\{i_{1},\ldots,i_{s}\\}\neq\\{j_{1},\ldots,j_{s}\\}$ such that
$a_{l(i_{1})}\cdots a_{l(i_{s})}=a_{l(j_{1})}\cdots a_{l(j_{s})}.$

###### Proof of Theorem 1.2.

Similarly as in the proof of Lemma 3.4, we may suppose that
$M=\mathbb{B}(R_{0})\subset{\mathbb{C}}^{m}$. Suppose contrarily that
$f\not\equiv g$.

Consider the simple graph $\mathcal{H}$, where the set of vertice is
$\\{1,2,\ldots,q\\}$ and the set of edges consists of all pair
$\\{(i,j)|\frac{H_{i}(f)}{H_{j}(f)}\not\equiv\frac{H_{i}(g)}{H_{j}(g)}\\}$.
Since $f\not\equiv g$, the degree of $\mathcal{H}$ at every vertex is at least
$q-n>\frac{q}{2}$. By Dirac’s theorem, $\mathcal{H}$ has a Hamiltonian cycle,
for instance it is $(1,2,3,\ldots,q,1)$. Therefore

$P_{i}=H_{i}(f)H_{\sigma(i)}(g)-H_{i}(g)H_{\sigma(i)}(f)\not\equiv 0,$

where $\sigma(i)=i+1$ for $i<q$ and $\sigma(q)=1$. We easily see that

$\nu_{P_{i}}(z)\geq\sum_{j=i,\sigma(i)}\min\left\\{\nu_{H_{j}(f)}(z),\nu_{H_{j}(g)}(z)\right\\}+\sum_{{\mathrel{\mathop{{j=1}}\limits_{{i\neq
j\neq\sigma(i)}}}}}^{q}\min\left\\{\nu_{H_{j}(f)}(z),1\right\\}$

for all $z$ ouside the analytic subset $\bigcup_{1\leq u<v\leq
q}f^{-1}(H_{u}\cap H_{v})$, which is of codimension two.

Define $\nu_{i}=\min\\{1,|\nu_{H_{i}(f)}-\nu_{H_{i}(g)}|\\}$ and
$\ell^{\prime}=\max\\{0,\ell-n\\}$. We see that if $\nu_{i}(z)\neq 0$ then
$\min\\{\nu_{H_{i}(f)},\nu_{H_{i}(g)}\\}\geq\ell$. Then, by setting
$P=\prod_{i=1}^{q}P_{i}\not\equiv 0$, we have

(4.5) $\displaystyle\begin{split}\nu_{P}&\geq
2\sum_{j=1}^{q}\min\left\\{\nu_{H_{j}(f)},\nu_{H_{j}(g)}\right\\}+(q-2)\sum_{j=1}^{q}\nu^{[1]}_{H_{j}(f)}\\\
&\geq
2\sum_{j=1}^{q}\left(\nu^{[n]}_{H_{j}(f)}+\nu^{[n]}_{H_{j}(g)}-n\nu^{[1]}_{H_{j}(f)}+\ell^{\prime}\nu_{i}\right)+(q-2)\sum_{j=1}^{q}\nu^{[1]}_{H_{j}(f)}\\\
&=2\sum_{j=1}^{q}(\nu^{[n]}_{H_{j}(f)}+\nu^{[n]}_{H_{j}(g)})+\frac{q-2n-2}{2}\sum_{j=1}^{q}(\nu^{[1]}_{H_{j}(f)}+\nu^{[1]}_{H_{j}(g)})+2\ell^{\prime}\sum_{j=1}^{q}\nu_{i}\\\
&\geq\frac{q+2n-2}{2n}\sum_{j=1}^{q}(\nu^{[n]}_{H_{j}(f)}+\nu^{[n]}_{H_{j}(g)})+2\ell^{\prime}\sum_{j=1}^{q}\nu_{i}.\end{split}$

It is clear that $P\in B\big{(}q,0;f,g)$. By Lemma 3.1, we have

$\displaystyle\left(\frac{q+2n-2}{2n}(q-n-1)-q\right)\cdot\left(\frac{1}{\rho_{f}}+\frac{1}{\rho_{g}}\right)\leq\frac{q+2n-2}{n}(\ell_{f}+\ell_{g})$
$\displaystyle\Rightarrow\hskip 30.0pt$
$\displaystyle\frac{q^{2}-(n+3)q-(n+1)(2n-2)}{q+2n-2}\leq\frac{2\rho_{f}\rho_{g}(\ell_{f}+\ell_{g})}{\rho_{f}+\rho_{g}}$
$\displaystyle\Rightarrow\hskip 30.0pt$
$\displaystyle\frac{q(q-2)}{q+2n-2}\leq n+1+\rho_{f,g}(\ell_{f}+\ell_{g}).$

This implies that

(4.6) $\displaystyle
q\leq\frac{q+2n-2}{q-2}\left(n+1+\rho_{f,g}(\ell_{f}+\ell_{g})\right).$

We now prove the theorem in the following two cases.

(a) Assume that $\ell=1$. Since $q\geq 2n+2$, one has $\frac{q+2n-2}{q-2}\leq
2$. Then, from (4.6) we get

$q\leq 2n+2+2\rho_{f,g}(\ell_{f}+\ell_{g}).$

This is a contradiction. Therefore, the supposition is false. Hence, $f\equiv
g$.

(b) Assume that $\ell>n$. Then $\ell^{\prime}>0$. Because of Part a), it is
enough for us to consider the case where $q\leq
2n+2+2\rho_{f,g}(\ell_{f}+\ell_{g}).$

We set $h_{i}=\frac{H_{i}(f)}{H_{i}(g)}\ (1\leq i\leq q).$ Then
$\frac{h_{i}}{h_{j}}=\frac{H_{i}(f)\cdot H_{j}(g)}{H_{j}(f)\cdot H_{i}(g)}$
does not depend on the choice of the reduced representations of $f$ and $g$
respectively.

Take an arbitrary subset of $2n+2$ elements of the set $\\{1,\ldots,q\\}$, for
instance it is $\\{1,\ldots,2n+2\\}$. Denote by $\mathcal{I}$ the set of all
combinations $I=(i_{1},\ldots,i_{n+1})$ with $1\leq i_{1}<...<i_{n+1}\leq q$.
For each $I=(i_{1},\ldots,i_{n+1})\in\mathcal{I}$, put
$h_{I}=\prod_{j=1}^{n+1}h_{i_{j}}$ and define

$A_{I}=(-1)^{\frac{(n+1)(n+2)}{2}+i_{1}+\cdots+i_{n+1}}\cdot\det(a_{i_{r}l};1\leq
r\leq n+1,0\leq l\leq n)$
$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\det(a_{j_{s}l};1\leq
s\leq n+1,0\leq l\leq n),$

where $J=(j_{1},\ldots,j_{n+1})\in\mathcal{I}$ such that $I\cup
J=\\{1,2,\ldots,2n+2\\}.$ Since

$\sum_{k=0}^{n}a_{ik}f_{k}-h_{i}\cdot\sum_{k=0}^{n}a_{ik}g_{k}=0\ (1\leq i\leq
2n+2),$

one has

$\det\ (a_{i0},\ldots,a_{in},a_{i0}h_{i},\ldots,a_{in}h_{i};1\leq i\leq
2n+2)=0.$

Therefore, $\sum_{I\in\mathcal{I}}A_{I}h_{I}=0$ (note that
$A_{I}\in{\mathbb{C}}^{*}$).

Take $I_{0}\in\mathcal{I}$. Denote by $t$ the minimal number satisfying the
following: There exist $t$ elements
$I_{1},\ldots,I_{t}\in\mathcal{I}\setminus\\{I_{0}\\}$ and $t$ nonzero
constants $b_{i}\in{\mathbb{C}}^{*}$ such that
$\sum_{i=0}^{t}b_{i}h_{I_{i}}=0.$ By the minimality of $t$, the family
$\\{h_{I_{1}},\ldots,h_{I_{t}}\\}$ is linearly independent over
${\mathbb{C}}$.

Case 1: $t=1$. Then $\dfrac{h_{I_{0}}}{h_{I_{1}}}\in{\mathbb{C}}^{*}.$

Case 2: $t\geq 2.$ Consider the linearly nondegenerate meromorphic mapping $F$
from $\mathbb{B}^{m}(R_{0})$ into ${\mathbb{P}}^{t-1}({\mathbb{C}})$ with a
reduced representation $F=(h_{I_{1}}d:\cdots:h_{I_{t}}d),$ where $d$ is a
meromorphic function. We see that

$\displaystyle\sum_{i=0}^{t}\nu_{dh_{I_{i}}}^{[1]}(z)\leq$
$\displaystyle\sum_{j=1}^{2n+2}\sharp\\{i|j\in
I_{i},\nu_{H_{j}(f)}(z)>\nu_{H_{j}(g)}(z)\\}$ $\displaystyle\ \ \
+\sum_{i=1}^{2n+2}\sharp\\{i|j\not\in
I_{i},\nu_{H_{j}(f)}(z)<\nu_{H_{j}(g)}(z)\\}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2n+2}\frac{p}{2}\nu_{i}(z)\leq\frac{p}{2}\sum_{i=1}^{q}\nu_{i}(z),$

for every $z$ outside an analytic set of codimension two. Here by $\sharp S$
we denote the number of elements of the set $S$.

It is clear that $T_{F}(r,r_{0})\leq(n+1)(T_{f}(r,r_{0})+T_{g}(r,r_{0}))$. Let
$W(F)$ be a generalized Wronskian of $F$ and set

$G:=\prod_{0\leq s<l\leq 2}\left(\dfrac{(h_{I_{l}}d-h_{I_{s}}d)\cdot
W(F)}{\prod_{i=0}^{t}(h_{I_{i}}d)}\right).$

Then we have $G\in B(0,3(t-1)(t+1)/2;F)\subset B(0,3(p-2)p/2;f,g)$. For each
subset $J\subset\\{1,\ldots,q\\}$, set $J^{c}=\\{1,\ldots,q\\}\setminus J$. It
is clear that

$\bigcup_{0\leq s<l\leq 2}((I_{l}\setminus I_{s})\cup(I_{s}\setminus
I_{l}))^{c}=\\{1,\ldots,q\\}.$

We have

$\displaystyle-\nu_{G}$
$\displaystyle=3\sum_{i=0}^{t}\nu_{h_{I_{i}}d}-3\nu_{W(F)}-\sum_{0\leq s<l\leq
2}\nu_{h_{I_{l}}d-h_{I_{s}}d}$ $\displaystyle\leq
3\sum_{i=0}^{t}\nu_{h_{I_{i}}d}^{[t-1]}-\sum_{0\leq s<l\leq
2}(\nu^{0}_{h_{I_{l}}/h_{I_{s}}-1})$ $\displaystyle\leq
3(t-1)\sum_{i=0}^{t}\nu_{h_{I_{i}}d}^{[1]}-\sum_{0\leq s<l\leq
2}\sum_{i\in((I_{l}\setminus I_{s})\cup(I_{s}\setminus
I_{l}))^{c}}\nu^{[1]}_{H_{i}(f)}$ $\displaystyle\leq
3(p-2)\sum_{i=0}^{t}\nu_{h_{I_{i}}d}^{[1]}-\sum_{i=1}^{q}\nu^{[1]}_{H_{i}(f)}\leq\frac{3(p-2)p}{2}\nu_{i}-\sum_{i=1}^{q}\nu^{[1]}_{H_{i}(f)}.$

Then, we have

$\displaystyle\nu_{\prod_{i=1}^{q}P_{i}}$
$\displaystyle\geq\frac{q+2n-2}{2n}\sum_{i=1}^{q}(\nu_{H_{i}(f)}^{[n]}+\nu_{H_{i}(g)}^{[n]})+2\ell^{\prime}\sum_{i=1}^{q}\nu_{i}$
$\displaystyle\geq\frac{q+2n-2}{2n}\sum_{i=1}^{q}\left(\nu_{H_{i}(f)}^{[n]}+\nu_{H_{i}(g)}^{[n]}\right)+\frac{4\ell^{\prime}}{3(p-2)p}\left(\sum_{i=1}^{q}\nu^{[1]}_{H_{i}(f)}-\nu_{G}\right).$

This yields that

$\nu_{G^{4\ell^{\prime}}(\prod_{i=1}^{q}P_{i})^{3(p-2)p}}\geq\left(3(p-2)p\frac{q+2n-2}{2n}+\frac{2\ell^{\prime}}{n}\right)\sum_{i=1}^{q}\left(\nu_{H_{i}(f)}^{[n]}+\nu_{H_{i}(g)}^{[n]}\right).$

Note that $G\in B(0,3(p-2)p/2;f,g)$ and $P_{i}\in B(1,0;f,g)$. Then
$G^{4\ell^{\prime}}(\prod_{i=1}^{q}P_{i})^{3(p-2)p}$ belongs to
$B(3q(p-2)p,6\ell^{\prime}(p-2)(p-1);f,g)$. From Lemma 3.1, we have

$\displaystyle q$ $\displaystyle\leq
n+1+\frac{3q(p-2)p}{3(p-2)p\frac{q+2n-2}{2n}+\frac{2\ell^{\prime}}{n}}+\rho_{f,g}\left(\ell_{f}+\ell_{g}+\frac{6\ell^{\prime}(p-2)(p-1)}{3(p-2)p\frac{q+2n-2}{2n}+\frac{2\ell^{\prime}}{n}}\right)$
$\displaystyle=n+1+\frac{3(2n+2+2\rho_{f,g}(\ell_{f}+\ell_{g}))}{3\frac{q+2n-2}{2n}+\frac{2\ell^{\prime}}{n(p-2)p}}+\rho_{f,g}\left(\ell_{f}+\ell_{g}+\frac{6\ell^{\prime}(p-2)(p-1)}{6(p-2)p+\frac{2\ell^{\prime}}{n}}\right)$
$\displaystyle\leq
n+1+\frac{3(2n+2)}{6+\frac{2\ell^{\prime}}{n(p-2)p}}+\rho_{f,g}\left(\ell_{f}+\ell_{g}+\frac{6\ell^{\prime}(p-2)(p-1)+2(\ell_{f}+\ell_{g})}{6(p-2)p+\frac{2\ell^{\prime}}{n}}\right)$
$\displaystyle=2n+1+\frac{3(p-2)p-\ell^{\prime}}{3(p-2)p+\ell^{\prime}/n}+\rho_{f,g}\left(\ell_{f}+\ell_{g}+\ell^{\prime}\right).$

This is a contradiction. Hence, this case does not happen.

Therefore, for each $I\in\mathcal{I}$, there is
$J\in\mathcal{I}\setminus\\{I\\}$ such that
$\dfrac{h_{I}}{h_{J}}\in{\mathbb{C}}^{*}.$

Consider the torsion free abelian subgroup generated by the family
$\\{[h_{1}],\ldots,[h_{q}]\\}$ of the abelian group
$\mathcal{M^{*}}_{m}/{\mathbb{C}}^{*}.$ Then the family
$\\{[h_{1}],\ldots,[h_{q}]\\}$ has the property $P_{q,n+1}$. By Proposition
4.4, there exist $q-2n\geq 2$ elements, without loss of generality we may
assume that they are $[h_{1}],[h_{i}]$ such that $[h_{1}]=[h_{i}].$ Then
$\dfrac{h_{1}}{h_{i}}=\lambda\in{\mathbb{C}}^{*}.$ Suppose that
$\lambda\not\equiv 1.$ Since $\dfrac{h_{1}(z)}{h_{i}(z)}=1$ for each
$z\in\bigcup_{{\mathrel{\mathop{{k\neq
i}}\limits^{k=2}}}}^{q}f^{-1}(H_{k})\setminus(f^{-1}(H_{1})\cup
f^{-1}(H_{i})),$ it implies that $\bigcup_{{\mathrel{\mathop{{k\neq
i}}\limits^{k=2}}}}^{q}f^{-1}(H_{k})=\emptyset$. Hence
$\sum_{{\mathrel{\mathop{{k\neq
i}}\limits^{k=2}}}}^{q}\nu_{H_{k}(f)}^{[n]}=\sum_{{\mathrel{\mathop{{k\neq
i}}\limits^{k=2}}}}^{q}\nu_{H_{k}(g)}^{[n]}=0$. Then, by Lemma 3.1, we have

$\displaystyle q-2\leq n+1+\rho_{f,g}(\ell_{f}+\ell_{g}).$

This is a contradiction. Thus, $\lambda\equiv 1$, i.e., $h_{1}\equiv h_{i}$.
Hence $\nu_{H_{1}(f)}=\nu_{H_{1}(g)}$ and $\nu_{H_{i}(f)}=\nu_{H_{i}(g)}$. By
the assumption, we note that $2<i<q$.

Now we consider

$\displaystyle P_{1}$
$\displaystyle=H_{1}(f)H_{2}(g)-H_{2}(f)H_{1}(g)=\frac{H_{1}(f)}{H_{i}(f)}\left(H_{i}(f)H_{2}(f)-H_{2}(f)H_{i}(g)\right)\not\equiv
0.$

From this inequality, we easily see that

(4.7)
$\displaystyle\nu_{P_{1}}\geq(\nu_{H_{1}(f)}+\nu_{H_{1}(f)}^{[1]})+\nu_{H_{2}(f)}^{[n]}+\sum_{k=3}^{q}\nu_{H_{k}(f)}^{[1]}$

and similarly

(4.8)
$\displaystyle\begin{split}\nu_{P_{q}}&\geq(\nu_{H_{1}(f)}+\nu_{H_{1}(f)}^{[1]})+\nu_{H_{q}(f)}^{[n]}+\sum_{k=2}^{q-1}\nu_{H_{k}(f)}^{[1]},\\\
\nu_{P_{i-1}}&\geq(\nu_{H_{i}(f)}+\nu_{H_{i}(f)}^{[1]})+\nu_{H_{i-1}(f)}^{[n]}+\sum_{{\mathrel{\mathop{{k=1}}\limits_{{k\neq
i-1,i}}}}}^{q}\nu_{H_{k}(f)}^{[1]},\\\
\nu_{P_{i}}&\geq(\nu_{H_{i}(f)}+\nu_{H_{i}(f)}^{[1]})+\nu_{H_{i+1}(f)}^{[n]}+\sum_{{\mathrel{\mathop{{k=1}}\limits_{{k\neq
i,i+1}}}}}^{q}\nu_{H_{k}(f)}^{[1]}.\\\ \end{split}$

Then, similar as (4.5), with the help of (4.7) and (4.8), we have

(4.9)
$\displaystyle\nu_{P}(z)\geq\frac{q+2n-2}{2n}\sum_{k=1}^{q}\left(\nu_{H_{k}(f)}^{[n]}(z)+\nu_{H_{k}(g)}^{[n]}(z)\right)+2\sum_{k=1,i}\nu_{H_{k}(f)}^{[1]}.$

Consider the simple graph $\mathcal{H}^{\prime}$, where the set of vertice is
$\\{1,\ldots,q\\}\setminus\\{1,i\\}$ and the set of edges consists of all
pairs $\\{u,v\\}$ such that
$\frac{H_{i}(u)}{H_{j}(v)}\not\equiv\frac{H_{u}(g)}{H_{v}(g)}$. Since
$f\not\equiv g$, the degree of $\mathcal{H}^{\prime}$ at every vertex is at
least $q-2-n\geq\frac{q-2}{2}$. By Dirac’s theorem, $\mathcal{H}^{\prime}$ has
a Hamiltonian cycle $j_{1},\ldots,j_{q-2},j_{1}$. Therefore,

$P_{u}^{\prime}=H_{j_{u}}(f)H_{\sigma^{\prime}(u)}(g)-H_{j_{u}}(g)H_{\sigma^{\prime}(u)}(f)\not\equiv
0,$

where $\sigma^{\prime}(u)=j_{u+1}$ for $u<q-2$ and
$\sigma^{\prime}(q-2)=j_{1}$. We easily see that

$\nu_{P^{\prime}_{u}}(z)\geq\sum_{k=u,\sigma^{\prime}(u)}\min\left\\{\nu_{H_{j_{u}}(f)}(z),\nu_{H_{j_{u}}(g)}(z)\right\\}+\sum_{{\mathrel{\mathop{{k=1}}\limits_{{k\neq
u,\sigma^{\prime}(u)}}}}}^{q-2}\nu^{[1]}_{H_{j_{u}}(f)}(z)+\sum_{k=1,i}\nu^{[1]}_{H_{k}(f)}(z)$

for all $z$ outside the analytic subset $\bigcup_{1\leq u<v\leq
q}f^{-1}(H_{u}\cap H_{v})$, which is of codimension two. Let
$P^{\prime}=\prod_{u=1}^{q-2}P_{u}^{\prime}$ and similar as (4.5), we get

$\displaystyle\nu_{P^{\prime}}(z)$
$\displaystyle\geq\frac{q+2n-4}{2n}\sum_{{\mathrel{\mathop{{k\neq
i}}\limits^{k=2}}}}^{q}\left(\nu_{H_{k}(f)}^{[n]}(z)+\nu_{H_{k}(g)}^{[n]}(z)\right)+(q-2)\sum_{k=1,i}\nu_{H_{k}(f)}^{[1]}(z)$
$\displaystyle=\frac{q+2n-4}{2n}\sum_{k=1}^{q}\left(\nu_{H_{k}(f)}^{[n]}(z)+\nu_{H_{k}(g)}^{[n]}(z)\right)-(2n-2)\sum_{k=1,i}\nu_{H_{k}(f)}^{[1]}(z).$

It is clear that $P^{n-1}P^{\prime}\in B(nq-2,0;f,g)$ and satisfying

$\nu_{P^{n-1}P^{\prime}}\geq\frac{n(q+2n-2)-2}{2n}\sum_{k=1}^{q}\left(\nu_{H_{k}(f)}^{[n]}(z)+\nu_{H_{k}(g)}^{[n]}(z)\right).$

Then from Lemma 3.1, we have

$\displaystyle q$ $\displaystyle\leq
n+1+\rho_{f,g}(\ell_{f}+\ell_{g})+\frac{2n(nq-2)}{n(q+2n-2)-2}$
$\displaystyle\leq
n+1+\rho_{f,g}(\ell_{f}+\ell_{g})+\frac{2n(2n^{2}+2n-2)}{4n^{2}-2}+\frac{4n^{2}\rho_{f,g}(\ell_{f}+\ell_{g})}{4n^{2}-2}$
$\displaystyle=2n+1+\frac{2n-1}{2n-1/n}+\rho_{f,g}\frac{4n^{2}-1}{2n^{2}-1}(\ell_{f}+\ell_{g}).$

This is a contradiction.

Hence, we must have $f\equiv g$. The theorem is proved. ∎

##  5\. algebraic dependence problem

######  Lemma 5.1 (See [11, Lemma 3.1]).

Let $f^{1},f^{2},\ldots,f^{k}$ be as in Theorem 1.3 and
$M=\mathbb{B}^{m}(R_{0})$. Assume that each $f^{u}$ has a reduced
representation $f^{u}=(f^{u}_{0}:\cdots:f^{u}_{n})$, $1\leq u\leq k$. Suppose
that there exist integers $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq q$ such that

$P:=\mathrm{det}\left(H_{i_{j}}(f^{u})\right)_{1\leq j,u\leq k}\not\equiv 0.$

Then we have

$\displaystyle\nu_{P}(z)\geq\sum_{j=1}^{k}\left(\min_{1\leq u\leq
k}\\{\nu_{H_{i_{j}}(f^{u})}(z)\\}-\nu^{[1]}_{H_{i_{j}}(f^{1})}(z)\right)+(k-1)\sum_{i=1}^{q}\nu^{[1]}_{H_{i}(f^{1})}(z),$

for every $z\in\mathbb{B}^{m}(R_{0})$ outside an analytic set of codimension
two.

###### Proof of Theorem 1.3.

As usual, we may suppose that $M=\mathbb{B}^{m}({\mathbb{R}}_{0})$. For each
$1\leq i\leq q,$ we put $V_{i}:=((f^{1},H_{i}),(f^{2},H_{i}),(f^{3},H_{i}))$.
We write $V_{i}\cong V_{j}$ if $V_{i}\wedge V_{j}\equiv 0$, otherwise we write
$V_{i}\not\cong V_{j}.$

Suppose that $f^{1}\wedge f^{2}\wedge f^{3}\not\equiv 0$. Without loss of
generality, we may assume that

$\underbrace{V_{1}\cong\cdots\cong V_{l_{1}}}_{\text{ group
}1}\not\cong\underbrace{V_{l_{1}+1}\cong\cdots\cong V_{l_{2}}}_{\text{ group
}2}\not\equiv\underbrace{V_{l_{2}+1}\cong\cdots\cong V_{l_{3}}}_{\text{ group
}3}\not\cong\cdots\not\cong\underbrace{V_{l_{s-1}}\cong\cdots\cong
V_{l_{s}}}_{\text{ group }s},$

where $l_{s}=q.$ For each $1\leq i\leq q$, we set

$\sigma(i)=\begin{cases}i+n&\text{ if }i+n\leq q,\\\ i+n-q&\text{ if
}i+n>q.\end{cases}$

Since each group has at most $n$ elements, $V_{i}$ and $V_{\sigma(i)}$ belong
to two distinct groups, i.e., $V_{i}\wedge V_{\sigma(i)}\not\equiv 0$. Then,
we may choose another index, denoted by $\gamma(i)$, such that

$V_{i}\wedge V_{\sigma(i)}\wedge V_{\gamma(i)}\not\equiv 0.$

We set

$P_{i}:=\det\left(H_{i}(f^{u}),H_{\sigma(i)}(f^{u}),H_{\gamma(i)}(f^{u});1\leq
u\leq 3\right)\not\equiv 0.$

Then, by Lemma 5.1 we have

$\displaystyle\nu_{P_{i}}$
$\displaystyle\geq\sum_{j=i,\sigma(i)}\left(\min_{1\leq u\leq
3}\nu_{H_{j}(f^{u})}-\nu^{[1]}_{H_{j}(f^{1})}\right)+2\sum_{j=1}^{q}\nu^{[1]}_{H_{j}(f^{1})}$
$\displaystyle\geq\sum_{j=i,\sigma(i)}\left(\sum_{u=1}^{3}\nu^{[n]}_{H_{j}(f^{u})}-(2n+1)\nu^{[1]}_{H_{j}(f^{1})}\right)+2\sum_{j=1}^{q}\nu^{[1]}_{H_{j}(f^{1})}.$

Summing-up both sides of the above inequality over all $1\leq i\leq q$, we
have

$\nu_{\prod_{i=1}^{q}P_{i}}\geq
2\sum_{u=1}^{3}\sum_{j=1}^{q}\nu^{[n]}_{H_{j}(f^{u})}+(2q-4n-2)\sum_{j=1}^{q}\nu^{[1]}_{H_{j}(f^{u})}\geq\frac{2q+2n-2}{3n}\sum_{u=1}^{3}\sum_{j=1}^{q}\nu^{[n]}_{H_{j}(f^{u})}.$

It is easy to see that $\prod_{i=1}^{q}P_{i}\in B(q,0;f^{1},f^{2},f^{3}).$
Then, by Lemma 3.1 we have

$\frac{2q+2n-2}{3n}(q-n-1)-q\leq
2\frac{2q+2n-2}{3n}\left(\sum_{u=1}^{3}\frac{1}{\rho_{f^{u}}}\right)^{-1}\sum_{u=1}^{3}\ell_{f^{u}},$

i.e.,

$q\leq
n+1+\frac{3nq}{2q+2n-2}+2\left(\sum_{u=1}^{3}\frac{1}{\rho_{f^{u}}}\right)^{-1}\sum_{u=1}^{3}\ell_{f^{u}}.$

This is a contradiction. Hence $f^{1}\wedge f^{2}\wedge f^{3}\equiv 0$. The
theorem is proved. ∎

###### Proof of Theorem 1.4.

Denote by $\mathcal{I}$ the set of all $k$-tuples
$I=(i_{1},\ldots,i_{k})\in\mathbb{N}^{k}$ with $1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq q$. Suppose contrarily that $f^{1}\times
f^{2}\times\cdots\times f^{k}$ is not algebraically degenerate. Then for every
$I=(i_{1},\ldots,i_{k})\in\mathcal{I}$,

$P_{I}:=\det\left(H_{i_{j}}(f^{u})\right)_{1\leq j,u\leq k}\not\equiv 0.$

By Lemma 5.1, we have

$\displaystyle\nu_{P_{I}}$ $\displaystyle\geq\sum_{j=1}^{k}\left(\min_{1\leq
u\leq
k}\nu_{H_{i_{j}}(f^{u})}-\nu^{[1]}_{H_{i_{j}}(f^{1})}\right)+(k-1)\sum_{i=1}^{q}\nu^{[1]}_{H_{i}(f)}$
$\displaystyle\geq\sum_{j=1}^{k}\left(\nu^{[n]}_{H_{i_{j}}(f^{1})}-\nu^{[1]}_{H_{i_{j}}(f^{1})}\right)+(k-1)\sum_{i=1}^{q}\nu^{[1]}_{H_{i}(f)}.$

Setting $P=\prod_{I\in\mathcal{I}}P_{I}$ and summing up both sides of the
above inequality over all $I\in\mathcal{I}$, we get

(5.2)
$\displaystyle\begin{split}\nu_{P}&\geq\sharp\mathcal{I}\cdot\sum_{i=1}^{q}\left(\frac{k}{q}\nu^{[n]}_{H(f^{1})}+\frac{((k-1)q-k)}{q}\nu^{[1]}_{H_{i_{j}}(f^{1})}\right)\\\
&=\sharp\mathcal{I}\cdot\left(\frac{1}{q}+\frac{((k-1)q-k)}{knq}\right)\sum_{u=1}^{k}\sum_{i=1}^{q}\nu^{[n]}_{H_{i}(f^{u})}.\end{split}$

Applying Lemma 3.1 for the function $P\in
B(\sharp\mathcal{I},0;f^{1},\ldots,f^{k})$, we get

$\sum_{u=1}^{k}\frac{\sharp\mathcal{I}\cdot\left(\frac{1}{q}+\frac{((k-1)q-k)}{knq}\right)(q-n-1)-\sharp\mathcal{I}}{\rho_{f^{u}}}-2\sum_{u=1}^{k}\sharp\mathcal{I}\cdot\left(\frac{1}{q}+\frac{((k-1)q-k)}{knq}\right)\ell_{f^{u}}\leq
0,$

i.e.,

$q\leq
n+1+\frac{knq}{(k-1)q+k(n-1)}+2\left(\sum_{u=1}^{k}\frac{1}{\rho_{f^{u}}}\right)^{-1}\sum_{u=1}^{k}\ell_{f^{u}}.$

This is a contradition. Therefore, $f^{1}\times\cdots\times f^{k}$ is
algebraically degenerate. The theorem is proved. ∎

Acknowledgements. This work was done during a stay of the author at the
Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to
thank the staff there, in particular the partially support of VIASM. This
research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 101.02-2021.12.

Disclosure statement. No potential conflict of interest was reported by the
author(s).

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Si Duc Quang
1Department of Mathematics, Hanoi National University of Education,
136-Xuan Thuy, Cau Giay, Hanoi, Vienam.
2Thang Long Institute of Mathematics and Applied Sciences
Nghiem Xuan Yem, Hoang Mai, Hanoi, Vietnam.
E-mail<EMAIL_ADDRESS>