# Holomorphic maps from complex discs intersecting hypersurfaces of projective
varieties

Si Duc Quang

###### Abstract.

In this paper, we establish second main theorems for holomorphic maps with
finite growth index on complex discs intersecting families of hypersurfaces
(moving and fixed) in projective varieties, where the small term is detailed
estimate for various cases. Our results are generalizations and extensions of
many previous second main theorems for holomorphic maps from ${\mathbb{C}}$
intersecting with hypersurfaces (moving and fixed) or moving hyperplanes.

††footnotetext:  2010 Mathematics Subject Classification: Primary 32H30;
Secondary 30D35, 32A22.
Key words and phrases: Second main theorem, holomorphic map, hypersurface,
finite growth index.

##  1\. Introduction

In 2020, M. Ru and N. Sibony [16] introduced the class of holomorphic maps
from complex discs $\Delta(R)=\\{z\in{\mathbb{C}};|z|<R\\}$ into projective
spaces with finite growth index, and then they proved some second main
theorems for such curves in the case where the targets are fixed values or
fixed hyperplanes. Using these results, some applications on the uniqueness
problem of holomorphic curves on discs have been given, such as [17, 5].

Recently, in [12] the author has established the second main theorem for the
case where the targets are slowly moving hyperplanes. This work is a
continuation of our studies in [12]. In this paper we will establish second
main theorems for holomorphic maps with finite growth index on $\Delta(R)$
intersecting families of hypersurfaces where the small term is detailed
estimated in the following cases:

  * •

The targets are arbitrary slowly moving hypersurfaces $\\{Q_{i}\\}_{i=1}^{q}$
in ${\mathbb{P}}^{n}({\mathbb{C}})$, the curve is algebraically nondegenerate
over the field $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$ generated by the
coefficients of the moving hypersurfaces, the truncation level $L_{0}$ for the
counting functions is explicitly estimated.

  * •

The targets are slowly moving hypersurfaces $\\{Q_{i}\\}_{i=1}^{q}$ in
${\mathbb{P}}^{n}({\mathbb{C}})$ in general position, the curve may be
algebraically degenerate, the truncation level for the counting function is
$N=\binom{n+d}{n}-1$, where $d=lcm(\deg Q_{1},\ldots,\deg Q_{q})$.

  * •

The targets are hypersurfaces $\\{Q_{i}\\}_{i=1}^{q}$ in subgeneral position
with respect to a projective variety $V\subset{\mathbb{P}}^{n}({\mathbb{C}})$,
the curve from $\Delta(R)$ into $V$ is nondegenerate over $I_{d}(V)$ (i.e.,
the image of $f$ is not contaned in any hypersurface $Q$ of degree $d$ with
$V\not\subset Q$), the truncation level for the counting function is
$H_{V}(d)-1$, where $H_{V}(d)$ is the Hilbert weight of $V$.

Our results will generalize and extend many results for the case of the curves
from ${\mathbb{C}}$, such as [18, 15, 3, 1, 9, 10, 11, 12]. We note that if
$R=+\infty$ or the growth index of the curve is zero then these generalization
are trivial and the proofs will not be changed. But for the general case where
$R<+\infty$ and the growth index of the curve is positive, the problem will be
more interesting since the error term in the second main theorem may not be
small term. Therefore, we have to carefully compute that small terms. Also,
the techniques for the case of the curves from ${\mathbb{C}}$ are not enough
for our purpose then we have to propose some new techniques.

In order to state the results we recall the following. Let $\nu$ be a divisor
on $\Delta(R)$. We consider $\nu$ as a function on $\Delta(R)$ with values in
$\mathbb{Z}$ such that $\mathrm{Supp}\,(\nu):=\\{z;\nu(z)\neq 0\\}$ is a
discrete subset of $\Delta(R)$. Let $k$ be a positive integer or $+\infty$.
The truncated counting function of $\nu$ is defined by:

$n^{[k]}(t)=\sum_{|z|\leq t}\min\\{k,\nu(z)\\}\ (0\leq t\leq R)\text{ and }\
N^{[k]}(r,\nu)=\int_{r_{0}}^{r}\dfrac{n^{[k]}(t)-n^{[k]}(0)}{t}dt.$

Here $r_{0}$ is a fixed positive number so that $0<r_{0}<R$. We will omit the
character [k] if $k=+\infty$.

Let $\varphi:\Delta(R)\rightarrow{\mathbb{C}}\cup\\{\infty\\}$ be a non-
constant meromorphic function. We denote by $\nu^{0}_{\varphi}$ (resp.
$\nu^{\infty}_{\varphi}$) the divisor of zeros (resp. divisor of poles) of
$\varphi$ and set $\nu_{\varphi}=\nu^{0}_{\varphi}-\nu^{\infty}_{\varphi}$. As
usual, we will write $N^{[k]}_{\varphi}(r)$ and $N^{[k]}_{1/\varphi}(r)$ for
$N^{[k]}(r,\nu^{0}_{\varphi})$ and $N^{[k]}(r,\nu^{\infty}_{\varphi})$
respectively.

Let $f:\Delta(R)\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a holomorphic
map with a reduced representation $\tilde{f}=(f_{0},\ldots,f_{n})$, where
$f_{0},\ldots,f_{n}$ are holomorphic functions on $\Delta(R)$ without common
zero. The characteristic function of $f$ (with respect to the Fubini-Study
form $\Omega_{n}$ on ${\mathbb{P}}^{n}({\mathbb{C}})$) is defined by

$T_{f}(r):=\int_{0}^{r}\frac{dt}{t}\int_{|z|<t}f^{*}\Omega_{n}.$

According to M. Ru-N. Sibony [16], the growth index of $f$ is defined by

$c_{f}=\mathrm{inf}\left\\{c>0\
\biggl{|}\int_{0}^{R}\mathrm{exp}(cT_{f}(r))dr=+\infty\right\\}.$

For the convenient, we will set $c_{f}=+\infty$ if $\left\\{c>0\
\bigl{|}\int_{0}^{R}\mathrm{exp}(cT_{f}(r))dr=+\infty\right\\}=\varnothing$.
Throughout this paper, we always assume that $c_{f}<+\infty$. A meromorphic
function $a$ on $\Delta(R)$ (which is regarded as a holomorphic map into
${\mathbb{P}}^{1}({\mathbb{C}})$) is said to be small with respect to $f$ if
$\|T_{a}(r)=o(T_{f}(r))$ as $r\rightarrow R$. Here (and throughout this
paper), the notation “$\|P$” means the assertion $P$ holds for all $r\in(0,R)$
outside a subset $S\subset(0,R)$ with
$\int_{S}e^{(1+\epsilon)(c_{f}+\epsilon)T_{f}(r)}dr<+\infty$ for some positive
number $\epsilon$. Also, we will write “$\|_{\gamma}P$” if $P$ holds for all
$r\in(0,R)$ outside a subset $E\subset(0,R)$ with
$\int_{E}\gamma(r)dr<+\infty$, where $\gamma(r)$ is a non-negative measurable
function defined on $(0,R)$ with $\int_{0}^{R}\gamma(r)dr=\infty$. We denote
by $\mathcal{K}_{f}$ the set of all meromorphic functions on $\Delta(R)$ small
with respect to $f$. It is easy to see that $\mathcal{K}_{f}$ is a field.

We denote by $\mathcal{H}$ the ring of all holomorphic functions on
$\Delta(R)$. Let $Q$ be a homogeneous polynomial in
$\mathcal{H}[x_{0},\dots,x_{n}]$ of degree $d\geq 1.$ Denote by $Q(z)$ the
homogeneous polynomial over $\mathbf{C}$ obtained by substituting a specific
point $z\in\Delta(R)$ into the coefficients of $Q$. We also call a moving
hypersurface in ${\mathbb{P}}^{n}(\mathbf{C})$ each homogeneous polynomial
$Q\in\mathcal{H}[x_{0},\dots,x_{n}]$ such that all coefficients of $Q$ have no
common zero. We denote by $Q(z)^{*}$ the support of the hypersurface $Q(z)$
for every point $z$.

Let $Q$ be a moving hypersurface in ${\mathbb{P}}^{n}(\mathbf{C})$ of degree
$d\geq 1$ given by

$Q(z)=\sum_{I\in\mathcal{T}_{d}}a_{I}\omega^{I},$

where $\mathcal{T}_{d}=\\{(i_{0},\ldots,i_{n})\in\mathbf{N}_{0}^{n+1}\ ;\
i_{0}+\cdots+i_{n}=d\\}$, $a_{I}\in\mathcal{H}$ and
$\omega^{I}=\omega_{0}^{i_{0}}\cdots\omega_{n}^{i_{n}}$ for
$I=(i_{0},\ldots,i_{n})$. We consider the holomorphic map
$Q^{\prime}:\Delta(R)\rightarrow{\mathbb{P}}^{N}(\mathbf{C})$, where
$N=\binom{n+d}{n}$, given by

$Q^{\prime}(z)=(a_{I_{0}}(z):\cdots:a_{I_{N}}(z))\
(\mathcal{T}_{d}=\\{I_{0},\ldots,I_{N}\\}).$

Here $I_{0}<\cdots<I_{N}$ in the lexicographic ordering. By changing the
homogeneous coordinates of ${\mathbb{P}}^{n}({\mathbb{C}})$ if necessary, we
may assume that for each given moving hypersurface as above,
$a_{I_{0}}\not\equiv 0$ (note that $I_{0}=(0,\ldots,0,d)$ and $a_{I_{0}}$ is
the coefficient of $\omega_{n}^{d}$). We set

$\tilde{Q}=\sum_{j=0}^{N}\frac{a_{I_{j}}}{a_{I_{0}}}\omega^{I_{j}}.$

We set $Q(\tilde{f})=\sum_{I\in\mathcal{T}_{d}}a_{I}f^{I}$ and
$\tilde{Q}(\tilde{f})=\sum_{j=0}^{N}\frac{a_{I_{j}}}{a_{I_{0}}}\tilde{f}^{I_{j}}$,
where $\tilde{f}^{I}=f_{0}^{i_{0}}\cdots f_{n}^{i_{n}}$ for
$I=(i_{0},\ldots,i_{n})$.

The moving hypersurface $Q$ is said to be “slow” (with respect to $f$) if $\|\
T_{Q^{\prime}}(r)=o(T_{f}(r))$. This is equivalent to $\|\
T_{\frac{a_{I_{j}}}{a_{I_{0}}}}(r)=o(T_{f}(r))$, i.e.,
$\frac{a_{I_{j}}}{a_{I_{0}}}\in\mathcal{K}_{f}$, for all $j=1,\ldots,N$.

Denote by $\mathcal{M}$ the field of all meromorphic functions on $\Delta(R)$.
Let $Q_{1},\ldots,Q_{q}$ be $q$ moving hypersurfaces in
${\mathbb{P}}^{n}(\mathbf{C})$, given by
$Q_{i}=\sum_{I\in\mathcal{T}_{d_{i}}}a_{iI}\omega^{I}.$ We denote by
$\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$ the smallest subfield of $\mathcal{M}$
which contains $\mathbf{C}$ and all $\frac{a_{iI}}{a_{iJ}}$ with
$a_{iJ}\not\equiv 0$. The family $\\{Q_{1},\ldots,Q_{q}\\}$ is said to be in
weakly $l-$subgeneral position if for every $1\leq j_{1}<\cdots<j_{l+1}\leq
q,$

$\bigcap_{s=1}^{l+1}Q_{s}(z)^{*}\cap V=\varnothing$

for generic points $z\in\Delta(R)$ (i.e., for all $z\in\Delta(R)$ outside a
discrete subset). Here, we note that $\dim\varnothing=-\infty$. In [13], we
define the distributive constant of $\\{Q_{i}\\}_{i=1}^{q}$ by

$\delta:={\mathrel{\mathop{{\max}}\limits_{{\Gamma\subset\\{1,\ldots,q\\}}}}}\dfrac{\sharp\Gamma}{n-\dim\left(\bigcap_{j\in\Gamma}Q_{j}(z)^{*}\right)}$

for generic points $z\in\Delta(R)$. From [13, Remark 3.7], we know that if
$\\{Q_{1},\ldots,Q_{q}\\}$ is in weakly $l-$subgeneral position then
$\delta\leq l-n+1$.

Our first purpose in this paper is to give a second main theorem for
holomorphic maps from $\Delta(R)$ into ${\mathbb{P}}^{n}({\mathbb{C}})$
intersecting an arbitrary family of slowly moving hypersurfaces with small
term being detailed estimated. Namely, we will prove the following.

######  Theorem 1.1.

Let $f$ be a nonconstant holomorphic map of $\Delta(R)$ into
${\mathbb{P}}^{n}(\mathbf{C})$ with finite growth index $c_{f}$. Let
$\\{Q_{i}\\}_{i=1}^{q}$ be an arbitrary family of slowly (with respect to $f$)
moving hypersurfaces with $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$ and with the
distributive constant $\delta$. Assume that $f$ is algebraically nondegenerate
over $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$. Then for any $\epsilon>0$, we have

$\|\
(q-\delta(n+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[L_{i}]}_{Q_{i}(f)}(r)+\frac{L_{0}(L_{0}+1)}{2}(\log\gamma(r)+\epsilon\log
r)+o(T_{f}(r)),$

where $\gamma(r)=e^{(1+\epsilon)(c_{f}+\epsilon)T_{f}(r)}$,
$L_{i}=\frac{1}{d_{i}}L_{0}$ and $L_{0}$ is a positive number which is defined
by:

$\displaystyle L_{0}$
$\displaystyle:=\binom{L+n}{n}p_{0}^{\binom{L+n}{n}\left(\binom{L+n}{n}-1\right)\binom{q}{n}-2}-1$
$\displaystyle\text{with }\ L$
$\displaystyle:=(n+1)d+2^{n+1}\delta(n+1)^{2}I(\epsilon^{-1})d,$
$\displaystyle d$ $\displaystyle:=lcm(d_{1},\ldots,d_{q})\text{(the least
common multiple of all $d_{i}$'s)},$ $\displaystyle\text{ and }\ p_{0}$
$\displaystyle:=[\dfrac{\binom{L+n}{n}(\binom{L+n}{n}-1)\binom{q}{n}-1}{\log(1+\frac{\epsilon}{3(n+1)\delta})}]^{2}.$

Here $I(\epsilon^{-1})$ denotes the smallest integer exceed $\epsilon^{-1}$,
$N^{[L]}_{Q(f)}(r)$ stands for the counting function with truncation level $L$
of the divisor of zeros of the function $Q(\tilde{f})$ for any hypersurface
$Q$ and a reduced representation $\tilde{f}$ of $f$. Note that, in this result
we do not need any condition on the position of the family of moving
hypersurfaces. This result is an extension of our recent second main theorem
in [13]. In this theorem, the constant $\delta(n+1)$ in front of $T_{f}(r)$ is
called the total defect.

Our second purpose in this paper is to establish the second main theorem with
better truncation level for the counting function. We will prove the
following.

######  Theorem 1.2.

Let $f$ be a nonconstant holomorphic map of $\Delta(R)$ into
${\mathbb{P}}^{n}(\mathbf{C})$ with finite growth index $c_{f}$. Let
$\\{Q_{i}\\}_{i=1}^{q}$ be a family of slowly (with respect to $f$) moving
hypersurfaces with $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$ in weakly general
position. Set $d=lcm(d_{1},\ldots,d_{q})$ and $N=\binom{n+d}{n}-1$. Assume
that $Q_{i}(f)\not\equiv 0\ (1\leq i\leq q)$ and $q\geq n(N-n+2)+1$. For every
$\epsilon>0$, we have

$\displaystyle\|\ T_{f}(r)\leq$
$\displaystyle\dfrac{N+2}{q-n(N-n+2)+N+1}\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[N]}_{Q_{i}(f)}(r)$
$\displaystyle+\frac{N(N+1)}{2}(\log\gamma(r)+\epsilon\log r)+o(T_{f}(r)),$

where $\gamma(r)=e^{(1+\epsilon)(c_{f}+\epsilon)T_{f}(r)}$.

Note that, in the above result we do not need “the condition on the degeneracy
of the mapping $f$”. With this condition, we will prove a better result as
follows.

######  Theorem 1.3.

Let $f$, $\\{Q_{i}\\}_{i=1}^{q}$ , $d$ and $N$ be as in Theorem 1.2. Assume
that $f$ is algebraically nondegenerate over
$\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$. For every $\epsilon>0$, we have

$\|\
T_{f}(r)\leq\dfrac{N+2}{q}\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[N]}_{Q_{i}(f)}(r)+\frac{N(N+1)}{2}(\log\gamma(r)+\epsilon\log
r)+o(T_{f}(r)),$

where $\gamma(r)=e^{(1+\epsilon)(c_{f}+\epsilon)T_{f}(r)}$.

Now, we consider the case where the holomorphic maps into a projective
subvariety of ${\mathbb{P}}^{n}({\mathbb{C}})$. Let $V$ be a complex
projective subvariety of $\mathbb{P}^{n}(\mathbb{C})$ of dimension $k\ (k\leq
n)$. Let $Q_{1},\ldots,Q_{q}\ (q\geq k+1)$ be $q$ hypersurfaces in
$\mathbb{P}^{n}(\mathbb{C})$. The family of hypersurfaces
$\\{Q_{i}\\}_{i=1}^{q}$ is said to be in $N$-subgeneral position with respect
to $V$ if for any $1\leq i_{1}<\cdots<i_{N+1}\leq q$,

$V\cap(\bigcap_{j=1}^{N+1}Q_{i_{j}}^{*})=\varnothing.$

Let $d$ be a positive integer. We denote by $I(V)$ the ideal of homogeneous
polynomials in $\mathbb{C}[x_{0},\ldots,x_{n}]$ defining $V$ and by $H_{d}$
the $\mathbb{C}$-vector space of all homogeneous polynomials in
$\mathbb{C}[x_{0},\ldots,x_{n}]$ of degree $d.$ Define

$I_{d}(V):=\dfrac{H_{d}}{I(V)\cap H_{d}}\text{ and }H_{V}(d):=\dim I_{d}(V).$

Then $H_{V}(d)$ is called the Hilbert function of $V$. The element of
$I_{d}(V)$ which is an equivalent class of a homogeneous polynomial $Q\in
H_{d},$ will be denoted by $[Q]$. A holomorphic map
$f:\Delta(R)\longrightarrow V$ is said to be degenerate over $I_{d}(V)$ if
there is no $[Q]\in I_{d}(V)\setminus\\{0\\}$ such that $Q(f)\equiv 0.$
Otherwise, we say that $f$ is nondegenerate over $I_{d}(V)$. It is clear that
if $f$ is algebraically nondegenerate, then $f$ is nondegenerate over
$I_{d}(V)$ for every $d\geq 1.$

Our last result is stated as follows.

######  Theorem 1.4.

Let $V$ be a projective subvariety of $\mathbb{P}^{n}(\mathbb{C})$ of
dimension $k\ (k\leq n)$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be hypersurfaces of
$\mathbb{P}^{n}(\mathbb{C})$ in $N$-subgeneral position with respect to $V$
with $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$ and $d=lcm(d_{1},\ldots,d_{q})$. Let
$f$ be a holomorphic map from $\Delta(R)$ into $V$ with finite growth index
$c_{f}$ such that $f$ is nondegenerate over $I_{d}(V)$. Then, for every
$\epsilon>0$ we have

$\displaystyle\|_{\gamma}\ $
$\displaystyle\left(q-\dfrac{(2N-k+1)H_{V}(d)}{k+1}\right)T_{f}(r)$
$\displaystyle\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[H_{V}(d)-1]}_{Q_{i}(f)}(r)+\frac{(H_{V}(d)-1)H_{V}(d)}{2}\left(\log\gamma+\epsilon\log
r\right)+o(T_{f}(r)),$

where $\gamma(r)=e^{(1+\epsilon)(c_{f}+\epsilon)T_{f}(r)}$.

##  2\. Proof of Theorem 1.1

Let $f$ be a holomorphic map from $\Delta(R)\ (0<R\leq+\infty)$ into
${\mathbb{P}}^{n}({\mathbb{C}})$. From now on, we always denote by $\tilde{f}$
a reduced representation of $f$ and $\tilde{f}=(f_{0},f_{1},\ldots,f_{n})$. We
set $\|\tilde{f}\|=(|f_{0}|^{2}+\cdots+|f_{n}|^{2})^{1/2}$.

In order to prove Theorem 1.1, we need the following results from [2, 3, 16,
9] and [13].

######  Theorem 2.1 (see [16, Theorem 4.8]).

Let $f$ be a linearly non-degenerate holomorphic map from $\Delta(R)\
(0<R\leq+\infty)$ into ${\mathbb{P}}^{n}({\mathbb{C}})$. Let $\gamma(r)$ be a
non-negative measurable function defined on $(0,R)$ with
$\int_{0}^{R}\gamma(r)dr=\infty$ and let $H_{1},\ldots,H_{q}$ be $q$ arbitrary
hyperplanes in ${\mathbb{P}}^{n}({\mathbb{C}})$. Then, for every
$\varepsilon>0$, we have

$\displaystyle\bigl{\|}_{\gamma}$
$\displaystyle\int_{0}^{2\pi}\max_{K}\sum_{j\in
K}\frac{\|\tilde{f}\|}{|H_{j}(\tilde{f})|}\frac{d\theta}{2\pi}+N_{W}(r)$
$\displaystyle\leq(n+1)T_{f}(r)+\frac{n(n+1)}{2}(\log\gamma(r)+\epsilon\log
r)+O(\log T_{f}(r)).$

Here $W$ is the generalized Wronskian of $f$, i.e., $W=\det(f_{i}^{(k)};0\leq
i,k\leq n)$ for a reduced representation $(f_{0},\ldots,f_{n})$ of $f$. We
also note that, in [16, Theorem 4.8] the authors require that $f$ is of finite
growth index (i.e., $c_{f}<+\infty$) and the obtained inequality is stated as
follows

$\displaystyle\bigl{\|}\ $ $\displaystyle\int_{0}^{2\pi}\max_{K}\sum_{j\in
K}\frac{\|\tilde{f}\|}{|H_{j}(\tilde{f})|}\frac{d\theta}{2\pi}+N_{W}(r)$
$\displaystyle\leq(n+1)T_{f}(r)+\frac{n(n+1)}{2}((1+\epsilon)(c_{f}+\epsilon)T_{f}(r)+\epsilon\log
r)+O(\log T_{f}(r)).$

Fortunately, in their proofs, once we keep using the function $\gamma(r)$
instead of substituting $\gamma(r)=e^{(c_{f}+\delta)T_{f}(r)}$ in the proof of
Lemma 4.3 and $\gamma(r)=e^{(c_{f}+\varepsilon)T_{f}(r)}$ in the proof of
Theorem 4.8 then we automatically get the second main theorem in the form of
Theorem 2.1.

As usual argument in the Nevanlinna theorem, from Theorem 2.1 we get the
following second main theorem.

######  Theorem 2.2 (see [16, Theorem 1.7]).

Let $f$ be a linearly non-degenerate holomorphic map from $\Delta(R)\
(0<R\leq+\infty)$ into ${\mathbb{P}}^{n}({\mathbb{C}})$. Let $\gamma(r)$ be a
non-negative measurable function defined on $(0,R)$ with
$\int_{0}^{R}\gamma(r)dr=\infty$ and let $H_{1},\ldots,H_{q}$ be $q$
hyperplanes in general position in ${\mathbb{P}}^{n}({\mathbb{C}})$. Then, for
every $\varepsilon>0$, we have

$\displaystyle\bigl{\|}_{\gamma}\ (q-n-1)T_{f}(r)\leq$
$\displaystyle\sum_{i=1}^{q}N^{[n]}_{H_{i}(f)}(r)+\frac{n(n+1)}{2}(1+\varepsilon)\log\gamma(r)$
$\displaystyle+O(\log T_{f}(r))+\frac{n(n+1)}{2}\varepsilon\log r.$

######  Lemma 2.3 (see [2, Lemma 2.2]).

Let $A$ be a commutative ring and let $\\{\phi_{1},\ldots,\phi_{p}\\}$ be a
regular sequence in $A$, i.e., for $i=1,\ldots,p,\phi_{i}$ is not a zero
divisor of $A/(\phi_{1},\ldots,\phi_{i-1})$. Denote by $I$ the ideal in $A$
generated by $\phi_{1},\ldots,\phi_{p}$. Suppose that for some
$q,q_{1},\ldots,q_{h}\in A$, we have an equation

$\phi_{1}^{i_{1}}\cdots\phi_{p}^{i_{p}}\cdot
q=\sum_{r=1}^{h}\phi_{1}^{j_{1}(r)}\cdots\phi_{p}^{j_{p}(r)}\cdot q_{r},$

where $(j_{1}(r),\ldots,j_{p}(r))>(i_{1},\ldots,i_{p})$ for $r=1,\ldots,h$.
Then $q\in I$.

Here, as throughout this paper, we use the lexicographic order on
$\mathbb{N}_{0}^{p}$. Namely,

$(j_{1},\ldots,j_{p})>(i_{1},\ldots,i_{p})$

iff for some $s\in\\{1,\ldots,p\\}$ we have $j_{l}=i_{l}$ for $l<s$ and
$j_{s}>i_{s}$.

######  Lemma 2.4 (see [3, Lemma 3.2]).

Let $\\{Q_{i}\\}_{i=1}^{q}\ (q\geq n+1)$ be a set of homogeneous polynomials
of common degree $d\geq 1$ in $\mathcal{K}_{f}[x_{0},\ldots,x_{n}]$ in weakly
general position. Then for any pairwise different $1\leq
j_{0},\ldots,j_{n}\leq q$ the sequence $\\{Q_{j_{0}},\ldots,Q_{j_{n}}\\}$ of
elements in $\mathcal{K}_{\\{Q_{i}\\}}[x_{0},\ldots,x_{n}]$ is a regular
sequence, as well as all its subsequences.

Let $Q_{1},\ldots,Q_{q}$ be $q$ moving hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$. From now on, denote simply by $\mathcal{K}$
the field $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$. Denote by
$\mathcal{C}_{\mathcal{K}}$ the set of all non-negative functions
$h:\Delta(R)\longrightarrow[0,+\infty]$, which are of the form

$h=\dfrac{|g_{1}|+\cdots+|g_{l}|}{|g_{l+1}|+\cdots+|g_{l+k}|},$

where $k,l\in\mathbf{N},\
g_{1},.\ldots,g_{l+k}\in\mathcal{K}\setminus\\{0\\}$. Also, we define the set
$\mathcal{C}_{f}$ similarly as $\mathcal{C}_{\mathcal{K}}$ where
${\mathcal{K}}$ is replaced by $\mathcal{K}_{f}$.

######  Lemma 2.5 (see [3]).

Let $\\{Q_{i}\\}_{i=0}^{n}$ be a set of homogeneous polynomials of degree $d$
in $\mathcal{K}_{f}[x_{0},...,x_{n}]$. Then there exists a function
$h_{1}\in\mathcal{C}_{f}$ such that

$\max_{i\in\\{0,...,n\\}}|Q_{i}(\tilde{f})|\leq h_{1}\|\tilde{f}\|^{d}.$

If, moreover, this set of homogeneous polynomials is in weakly general
position, then there exists a nonzero function $h_{2}\in\mathcal{C}_{f}$ such
that

$h_{2}\|\tilde{f}\|^{d}\leq\max_{i\in\\{0,...,n\\}}|Q_{i}(\tilde{f})|.$

######  Lemma 2.6 (see [13, Lemma 3.1]).

Let $Q_{1},\ldots,Q_{l}$ be $l$ hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$ of the same degree $d\geq 1$, such that
$\bigcap_{i=1}^{l}Q_{i}^{*}=\varnothing$ and

$\dim\left(\bigcap_{i=1}^{s}Q_{i}^{*}\right)=n-u\ \forall t_{u}\leq
s<t_{u+1},1\leq u\leq n,$

where $t_{1},t_{2},\ldots,t_{n}$ are integers with
$1=t_{1}<t_{2}<\cdots<t_{n+1}=l$. Then there exist $n+1$ hypersurfaces
$P_{1},\ldots,P_{n+1}$ in ${\mathbb{P}}^{n}({\mathbb{C}})$ of the forms

$P_{u}=\sum_{j=1}^{t_{u}}c_{uj}Q_{j},\ c_{uj}\in{\mathbb{C}},\
u=1,\ldots,n+1,$

such that $\bigcap_{u=1}^{n+1}P_{u}^{*}=\varnothing.$

######  Lemma 2.7 (see [13, Lemma 3.2]).

Let $t_{1},t_{2},\ldots,t_{n+1}$ be $n+1$ integers such that
$t_{1}<t_{2}<\cdots<t_{n+1}$, and let
$\delta={\mathrel{\mathop{{\max}}\limits_{{2\leq s\leq
n+1}}}}\dfrac{t_{s}-t_{1}}{s-1}$. Then for every $n$ real numbers
$a_{1},a_{2},\ldots,a_{n}$ with $a_{1}\geq a_{2}\geq\cdots\geq a_{n}\geq 1$,
we have

$a_{1}^{t_{2}-t_{1}}a_{2}^{t_{3}-t_{2}}\cdots
a_{n}^{t_{n+1}-t_{n}}\leq(a_{1}a_{2}\cdots a_{n})^{\delta}.$

###### Proof of Theorem 1.1.

Replacing $Q_{i}$ by $Q_{i}^{d/d_{i}}$ if necessary with the note that

$\dfrac{1}{d}N^{[L_{0}]}_{Q_{i}^{d/d_{i}}(f)}(r)\leq\frac{1}{d_{i}}N^{[L_{i}]}_{Q_{i}(f)}(r),$

we may assume that all hypersurfaces $Q_{i}\ (1\leq i\leq q)$ are of the same
degree $d$. We may also assume that $q>\delta(n+1)$.

Consider a reduced representation
$\tilde{f}=(f_{0},\ldots,f_{n}):\Delta(R)\rightarrow{\mathbb{C}}^{n+1}$ of
$f$. We also note that

$N^{[L_{0}]}_{Q_{i}(\tilde{f})}(r)=N^{[L_{0}]}_{\tilde{Q}_{i}(\tilde{f})}(r)+o(T_{f}(r)).$

Then without loss of generality we may assume that
$Q_{i}\in\mathcal{K}_{f}[x_{0},\ldots,x_{n}]$.

Take $z_{0}$ be a point such that

$\delta=\max_{\Gamma\subset\\{1,\ldots,q\\}}\dfrac{\sharp\Gamma}{n-\dim\bigcap_{j\in\Gamma}Q_{i}(z_{0})^{*}}.$

Note that $\delta\geq 1$, and hence $q>n+1$. If there exists
$i\in\\{1,\ldots,q\\}$ such that
$\bigcap_{{\mathrel{\mathop{{j=1}}\limits_{{j\neq
i}}}}}^{q}Q_{j}(z_{0})^{*}\neq\varnothing$ then

$\delta\geq\dfrac{q-1}{n}>\dfrac{q}{n+1}.$

This is a contradiction. Therefore,
$\bigcap_{{\mathrel{\mathop{{j=1}}\limits_{{j\neq
i}}}}}^{q}Q_{j}(z_{0})^{*}=\varnothing$ for all $i\in\\{1,2,\ldots,q\\}$.

Since the number of slowly moving hypersurfaces occurring in this proof is
finite, we may choose a function $c\in\mathcal{C}_{\mathcal{K}}$ such that for
each given slowly moving hypersurface $Q$ in this proof, we have

$Q(z)({\bf x})\leq c(z)\|{\bf x}\|^{\deg Q}$

for all ${\bf x}=(x_{0},\ldots,x_{n})\in{\mathbb{C}}^{n+1}$, $z\in\Delta(R)$.

We denote by $\mathcal{I}$ the set of all permutation $I=(i_{1},\ldots,i_{q})$
of the set $\\{1,\ldots,q\\}$. We set $n_{0}=\sharp\mathcal{I}$, $n_{0}=q!$
and write $\mathcal{I}=\\{I_{1},\ldots,I_{n_{0}}\\}$, where
$I_{i}=(I_{i}(1),\ldots,I_{i}(q))\in\mathbb{N}^{q}$ and
$I_{1}<I_{2}<\cdots<I_{n_{0}}$ in the lexicographic order.

For each $I_{i}\in\mathcal{I}$, since
$\bigcap_{j=1}^{q-1}Q_{I_{i}(j)}(z_{0})=\varnothing$, there exist $n+1$
integers $t_{i,0},t_{i,1},\ldots,t_{i,n}$ with
$0=t_{i,0}<\cdots<t_{i,n}=l_{i}$, where $l_{i}\leq q-2$ such that
$\bigcap_{j=0}^{l_{i}}Q_{I_{i}(j)}(z_{0})^{*}=\varnothing$ and

$\dim\left(\bigcap_{j=0}^{s}Q_{I_{i}(j)}(z_{0})^{*}\right)=n-u\ \forall
t_{i,u-1}\leq s<t_{i,u},1\leq u\leq n.$

Then, $\delta\geq\dfrac{t_{i,u}-t_{i,0}}{u}$ for all $1\leq u\leq n.$ Denote
by $P^{\prime}_{i,0},\ldots,P^{\prime}_{i,n}$ the hypersurfaces obtained in
Lemma 2.6 with respect to the hypersurfaces
$Q_{I_{i}(0)}(z_{0}),\ldots,Q_{I_{i}(l_{i})}(z_{0})$. Now, for each
$P^{\prime}_{i,j}$ constructed by

$P^{\prime}_{i,j}=\sum_{s=0}^{t_{i,j}}a_{i,j,s}Q_{I_{i}(s)}(z_{0})\
(a_{i,j,s}\in{\mathbb{C}})$

we define

$P_{i,j}(z)=\sum_{s=0}^{t_{i,j}}a_{i,j,s}Q_{I_{i}(s)}(z).$

Hence $\\{P_{i,j}\\}_{j=0}^{n}$ is a family of moving hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{C}})$ with $P_{i,j}(z_{0})=P^{\prime}_{i,j}$. Then
$\bigcap_{j=0}^{n}P_{i,j}(z_{0})^{*}=\varnothing$, and hence
$\\{P_{i,j}(z)\\}_{j=0}^{n}$ is in weakly general position.

We may choose a positive constant $b\geq 1$, commonly for all
$I_{i}\in\mathcal{I}$, such that

$|P_{i,j}(z)({\bf x})|\leq b\max_{0\leq s\leq t_{i,j}}|Q_{I_{i}(j)}(z)({\bf
x})|,$

for all $0\leq j\leq n$ and for all ${\bf
x}=(x_{0},\ldots,x_{n})\in{\mathbb{C}}^{n+1}$ and $z\in\Delta(R)$. Denote by
$\mathcal{S}$ the set of all points $z\in\Delta(R)$ such that
$\bigcap_{j=0}^{n}P_{i,j}(z)\neq\varnothing$ for some $I_{i}$. Then
$\mathcal{S}$ is a discrete subset of $\Delta(R)$.

By Lemma 2.5, there exists a function $A\in\mathcal{C}_{\mathcal{K}}$ such
that

$\|\tilde{f}(z)\|^{d}\leq A(z)\max_{0\leq j\leq
l_{i}}|Q_{I_{i}(j)}(z)(\tilde{f}(z))|\ (\forall I_{i}\in\mathcal{I}).$

For a point
$z\in\Delta\setminus\biggl{\\{}\bigcup_{i=1}^{q}Q_{i}(\tilde{f})^{-1}(\\{0\\})\cup\bigcup_{{\mathrel{\mathop{{0\leq
j\leq
n}}\limits_{{I_{i}\in\mathcal{I}}}}}}P_{i,j}(\tilde{f})^{-1}(\\{0\\})\biggl{\\}}$,
taking $I_{i}\in\mathcal{I}$ so that

$|Q_{I_{i}(0)}(z)(\tilde{f}(z))|\leq|Q_{I_{i}(1)}(z)(\tilde{f}(z))|\leq\cdots\leq|Q_{I_{i}(q-1)}(z)(\tilde{f}(z))|,$

by Lemma 3.4 we have

$\displaystyle\prod_{i=1}^{q}\dfrac{\|\tilde{f}(z)\|^{d}}{|Q_{i}(z)(\tilde{f}(z))|}$
$\displaystyle\leq\dfrac{A(z)^{q-l_{i}}}{c(z)^{l_{j}}}\prod_{j=0}^{l_{j}-1}\dfrac{c(z)\|\tilde{f}(z)\|^{d}}{|Q_{I_{i}(j)}(z)(\tilde{f}(z))|}$
$\displaystyle\leq\dfrac{A(z)^{q-l_{j}}}{c(z)^{l_{j}}}\prod_{j=0}^{n-1}\left(\dfrac{c(z)\|\tilde{f}(z)\|^{d}}{|Q_{I_{i}(t_{j})}(z)(\tilde{f}(z))|}\right)^{t_{i,j+1}-t_{i,j}}$
$\displaystyle\leq\dfrac{A(z)^{q-l_{j}}}{c(z)^{l_{j}}}\prod_{j=0}^{n-1}\left(\dfrac{c(z)\|\tilde{f}(z)\|^{d}}{|Q_{I_{i}(t_{j})}(z)(\tilde{f}(z))|}\right)^{\delta}$
$\displaystyle\leq\dfrac{A(z)^{q-l_{j}}b^{n\delta}}{c(z)^{l_{j}-n\delta}}\prod_{j=0}^{n-1}\left(\dfrac{\|\tilde{f}(z)\|^{d}}{|P_{i,j}(z)(\tilde{f}(z))|}\right)^{\delta}$
$\displaystyle\leq
C(z)\prod_{j=0}^{n}\left(\dfrac{\|\tilde{f}(z)\|^{d}}{|P_{i,j}(z)(\tilde{f}(z))|}\right)^{\delta},$

where $C\in\mathcal{C}_{\mathcal{K}}$.

Now, for each non-negative integer $L$, we denote by $V_{L}$ the vector space
(over $\mathcal{K}$) consisting of all homogeneous polynomials of degree $L$
in $\mathcal{K}[x_{0},\ldots,x_{n}]$ and the zero polynomial. Denote by
$(P_{i,1},\ldots,P_{i,n})$ the ideal in $\mathcal{K}[x_{0},\ldots,x_{n}]$
generated by $P_{i,1},\ldots,P_{i,n}$.

######  Lemma 2.8 (see [1, Lemma 5], [3, Proposition 3.3]).

Let $\\{P_{i}\\}_{i=1}^{q}\ (q\geq n+1)$ be a set of homogeneous polynomials
of common degree $d\geq 1$ in $\mathcal{K}_{f}[x_{0},\ldots,x_{n}]$ in weakly
general position. Then for any nonnegative integer $N$ and for any
$J:=\\{j_{1},\ldots,j_{n}\\}\subset\\{1,\ldots,q\\},$ the dimension of the
vector space $\frac{V_{L}}{(P_{j_{1}},\ldots,P_{j_{n}})\cap V_{L}}$ is equal
to the number of $n$-tuples $(s_{1},\ldots,s_{n})\in\mathbb{N}^{n}_{0}$ such
that $s_{1}+\cdots+s_{n}\leq L$ and $0\leq s_{1},\ldots,s_{n}\leq d-1$. In
particular, for all $L\geq n(d-1)$, we have

$\dim\frac{V_{L}}{(P_{j_{1}},\ldots,P_{j_{n}})\cap V_{L}}=d^{n}.$

For each positive integer $L$ divisible by $d$ and for each
$({\mathbf{i}})=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n}_{0}$ with
$\sigma({\mathbf{i}})=\sum_{s=1}^{n}i_{s}\leq\frac{L}{d}$, we set

$W^{I}_{({\mathbf{i}})}=\sum_{({\mathbf{j}})=(j_{1},\ldots,j_{n})\geq({\mathbf{i}})}P_{i,1}^{j_{1}}\cdots
P_{i,n}^{j_{n}}\cdot V_{L-d\sigma({\mathbf{j}})}.$

Hence $W^{I}_{(0,\ldots,0)}=V_{L}$ and $W^{I}_{({\mathbf{i}})}\supset
W^{I}_{({\mathbf{j}})}$ if $({\mathbf{i}})<({\mathbf{j}})$ in the
lexicographic ordering. Then $W^{I}_{({\mathbf{i}})}$ is a filtration of
$V_{L}$.

Let
$({\mathbf{i}})=(i_{1},\ldots,i_{n}),({\mathbf{i}}^{\prime})=(i_{1}^{\prime},\ldots,i_{n}^{\prime})\in\mathbb{N}^{n}_{0}$.
Suppose that $({\mathbf{i}}^{\prime})$ follows $({\mathbf{i}})$ in the
lexicographic ordering. As (3.5) in [10], we have

(2.9)
$\displaystyle\dim\dfrac{W^{I}_{({\mathbf{i}})}}{W^{I}_{({\mathbf{i}}^{\prime})}}=\dim\dfrac{V_{L-d\sigma({\mathbf{i}})}}{(P_{i,1},\ldots,P_{i,n})\cap
V_{L-d\sigma({\mathbf{i}})}}.$

Fix $L$ large enough (chosen later) and set $u=u_{L}:=\dim
V_{L}=\binom{L+n}{n}$. We assume that

$V_{L}=W^{I}_{({\mathbf{i}}_{1})}\supset
W^{I}_{({\mathbf{i}}_{2})}\supset\cdots\supset W^{I}_{({\mathbf{i}}_{K})},$

where $W^{I}_{({\mathbf{i}}_{s+1})}$ follows $W^{I}_{({\mathbf{i}}_{s})}$ in
the ordering and $({\mathbf{i}}_{K})=(\frac{L}{d},0,\ldots,0)$. It is easy to
see that $K$ is the number of $n$-tuples $(i_{1},\ldots,i_{n})$ with
$i_{j}\geq 0$ and $i_{1}+\cdots+i_{n}\leq\frac{L}{d}$. Then we have

$K=\binom{\frac{L}{d}+n}{n}.$

For each $k\in\\{1,\ldots,K-1\\}$ we set
$m^{I}_{k}=\dim\frac{W^{I}_{({\mathbf{i}}_{k})}}{W^{I}_{({\mathbf{i}}_{k+1})}}$,
and set $m^{I}_{K}=1$. By Lemma 2.8, $m^{I}_{k}$ does not depend on
$\\{P_{i,1},\ldots,P_{i,n}\\}$ and $k$, but on $\sigma({\mathbf{i}_{k}})$.
Hence, we set $m_{k}=m^{I}_{k}$. We also note that

(2.10) $\displaystyle m_{k}=d^{n}$

for all $k$ with $L-d\sigma({\mathbf{i}}_{k})\geq nd$ (it is equivalent to
$\sigma({\mathbf{i}}_{k})\leq\dfrac{L}{d}-n$).

Using the above filtration, we may choose a basis
$\\{\psi^{I}_{1},\cdots,\psi^{I}_{u}\\}$ of $V_{L}$ such that

$\\{\psi_{u-(m_{s}+\cdots+m_{K})+1},\ldots,\psi^{I}_{u}\\}$

is a basis of $W^{I}_{({\mathbf{i}}_{s})}$.

For each integer $l\ (0\leq l\leq\frac{L}{d})$, we set $m(l)=m_{k}$, where $k$
is an index such that $\sigma({\mathbf{i}}_{k})=l$. Since $m_{k}$ only depends
on $\sigma({\mathbf{i}}_{k})$, the above definition of $m(l)$ is well defined.
We see that

$\sum_{k=1}^{K}m_{k}i_{sk}=\sum_{l=0}^{\frac{L}{d}}\sum_{k|\sigma({\mathbf{i}}_{k})=l}m(l)i_{sk}=\sum_{l=0}^{\frac{L}{d}}m(l)\sum_{k|\sigma({\mathbf{i}}_{k})=l}i_{sk}.$

Here ${\mathbf{i}}_{k}=(i_{1k},\ldots,i_{nk})$. Note that, by the symmetry
$(i_{1},\ldots,i_{n})\rightarrow(i_{\sigma(1)},\ldots,i_{\sigma(n)})$ with
$\sigma\in S(n)$, $\sum_{k|\sigma({\mathbf{i}}_{k})=l}i_{sk}$ does not depend
on $s$. We set

$a:=\sum_{k=1}^{K}m_{k}i_{sk},\ \text{ which is independent of $s$ and $I$}.$

Hence, we have

$\displaystyle\log\prod_{l=1}^{u}|\psi^{I}_{l}(\tilde{f})(z)|$
$\displaystyle\leq
a\left(\log\prod_{j=1}^{n}\dfrac{|P_{i,j}(\tilde{f})(z)|}{\|\tilde{f}(z)\|^{d}}\right)+uL\log\|\tilde{f}(z)\|+\log
c_{I}(z),$

i.e.,

$\displaystyle
a\left(\log\prod_{j=1}^{n}\dfrac{\|\tilde{f}(z)\|^{d}}{|P_{i,j}(\tilde{f})(z)|}\right)\leq\log\prod_{l=1}^{u}\frac{\|\tilde{f}(z)\|^{L}}{|\psi^{I}_{l}(\tilde{f})(z)|}+\log
c_{I}(z),$

Set $c_{0}(z)=C(z)\prod_{I}(1+c_{I}^{\delta/a})\in\mathcal{C}_{f}$. Combining
the above inequality with (2.6), we obtain that

(2.11)
$\displaystyle\log\prod_{i=1}^{q}\dfrac{\|\tilde{f}(z)\|^{d}}{|Q_{i}(\tilde{f})(z)|}\leq\frac{\delta}{a}\log\prod_{l=1}^{u}\dfrac{\|\tilde{f}(z)\|^{L}}{|\psi^{I}_{l}(\tilde{f})(z)|}+\log
c_{0}.$

We now write

$\psi^{I}_{l}=\sum_{J\in\mathcal{T}_{L}}c^{I}_{lJ}x^{J}\in V_{L},\ \
c^{I}_{lJ}\in\mathcal{K}_{\\{Q_{i}\\}},$

where $\mathcal{T}_{L}$ is the set of all $(n+1)$-tuples
$J=(i_{0},\ldots,i_{n})$ with $\sum_{s=0}^{n}j_{s}=L$,
$x^{J}=x_{0}^{j_{0}}\cdots x_{n}^{j_{n}}$ and $l\in\\{1,\ldots,u\\}$. For each
$l$, we fix an index $J^{I}_{l}\in J$ such that $c^{I}_{lJ^{I}_{l}}\not\equiv
0$. Define

$\mu^{I}_{lJ}=\dfrac{c^{I}_{lJ}}{c^{I}_{lJ^{I}_{l}}},\ J\in\mathcal{T}_{L}.$

Set $\Phi=\\{\mu^{I}_{lJ};I\subset\\{1,\ldots,q\\},\sharp I=n,1\leq l\leq
M,J\in\mathcal{T}_{L}\\}$. Let $B=\sharp\Phi$ and note that $1\in\Phi$. One
has $B\leq
u\binom{q}{n}(\binom{L+n}{n}-1)=\binom{L+n}{n}(\binom{L+n}{n}-1)\binom{q}{n}$.
For each positive integer $l$, we denote by $\mathcal{L}(\Phi(l))$ the linear
span over ${\mathbb{C}}$ of the set

$\Phi(l)=\\{\gamma_{1}\cdots\gamma_{l};\gamma_{i}\in\Phi\\}.$

We see that

$\dim\mathcal{L}(\Phi(l))\leq\sharp\Phi(l)\leq\binom{B+l-1}{B-1}.$

We may choose a positive integer $p$ such that

$p\leq
p_{0}:=\left[\dfrac{B-1}{\log(1+\frac{\epsilon}{3(n+1)\delta})}\right]^{2}\text{
and }\dfrac{\dim\mathcal{L}(\Phi(p+1))}{\dim\mathcal{L}(\Phi(p))}\leq
1+\dfrac{\epsilon}{3(n+1)\delta}.$

Indeed, if
$\dfrac{\dim\mathcal{L}(\Phi(p+1))}{\dim\mathcal{L}(\Phi(p))}>1+\dfrac{\epsilon}{3(n+1)\delta}$
for all $p\leq p_{0}$, we have

$\dim\mathcal{L}(\Phi(p_{0}+1))\geq(1+\dfrac{\epsilon}{3(n+1)\delta})^{p_{0}}.$

Therefore, we have

$\displaystyle\log(1+\dfrac{\epsilon}{3(n+1)\delta})$
$\displaystyle\leq\dfrac{\log\dim\mathcal{L}(\Phi(p_{0}+1))}{p_{0}}\leq\dfrac{\log\binom{B+p_{0}}{B-1}}{p_{0}}$
$\displaystyle=\dfrac{1}{p_{0}}\log\prod_{i=1}^{B-1}\dfrac{p_{0}+i+1}{i}<\dfrac{(B-1)\log(p_{0}+2)}{p_{0}}$
$\displaystyle\leq\dfrac{B-1}{\sqrt{p_{0}}}\leq\dfrac{(B-1)\log(1+\frac{\epsilon}{3(n+1)\delta})}{B-1}$
$\displaystyle=\log(1+\frac{\epsilon}{3(n+1)\delta}).$

This is a contradiction.

We fix a positive integer $p$ satisfying the above condition. Put
$s=\dim\mathcal{L}(\Phi(p))$ and $t=\dim\mathcal{L}(\Phi(p+1))$. Let
$\\{b_{1},\ldots,b_{t}\\}$ be an ${\mathbb{C}}$-basis of
$\mathcal{L}(\Phi(p+1))$ such that $\\{b_{1},\ldots,b_{s}\\}$ be a
${\mathbb{C}}$-basis of $\mathcal{L}(\Phi(p))$.

For each $l\in\\{1,\ldots,u\\}$, we set

$\tilde{\psi}^{I}_{l}=\sum_{J\in\mathcal{T}_{L}}\mu^{I}_{lJ}x^{I}.$

For each $J\in\mathcal{T}_{L}$, we consider homogeneous polynomials
$\phi_{J}(x_{0},\ldots,x_{n})=x^{J}$. Let $F$ be a holomorphic mapping of
$\Delta(R)$ into ${\mathbb{P}}^{tu-1}({\mathbb{C}})$ with a reduced
representation $\tilde{F}=(hb_{i}\phi_{J}(\tilde{f}))_{1\leq i\leq
t,J\in\mathcal{T}_{L}}$, where $h$ is a nonzero meromorphic function on
$\Delta(R)$. We see that

$\|\ N_{h}(r)+N_{1/h}(r)=o(T_{f}(r)).$

Since $f$ is algebraically nondegenerate over $\mathcal{K}$, $F$ is linear
nondegenerate over ${\mathbb{C}}$. We see that there exist nonzero functions
$c_{1},c_{2}\in\mathcal{C}_{f}$ such that

$c_{1}|h|.\|\tilde{f}\|^{L}\leq\|\tilde{F}\|\leq c_{2}|h|.\|\tilde{f}\|^{L}.$

For each $l\in\\{1,\ldots,u\\},1\leq i\leq s$, we consider the linear form
$L^{I}_{il}$ in $x^{J}$ such that

$hb_{i}\tilde{\psi}^{I}_{l}(\tilde{f})=L^{I}_{il}(\tilde{F}).$

It is clear that $\\{b_{i}\tilde{\psi}^{I}_{l}(\tilde{f});1\leq i\leq s,1\leq
l\leq M\\}$ is linearly independent over ${\mathbb{C}}$, and so is
$\\{L^{I}_{il}(\tilde{F});1\leq i\leq s,1\leq l\leq u\\}$. This yields that
$\\{L^{I}_{il};1\leq i\leq s,1\leq l\leq u\\}$ is linearly independent over
${\mathbb{C}}$.

For every point $z$ which is neither zero nor pole of any
$hb_{i}\psi^{I}_{l}(\tilde{f})$, we also see that

$\displaystyle
s\log\prod_{l=1}^{u}\dfrac{\|\tilde{f}(z)\|^{L}}{|\psi^{I}_{l}(\tilde{f})(z)|}$
$\displaystyle=\log\prod_{{\mathrel{\mathop{{1\leq i\leq s}}\limits^{1\leq
l\leq u}}}}\dfrac{\|\tilde{F}(z)\|}{|hb_{i}\psi^{I}_{l}(\tilde{f})(z)|}+\log
c_{3}(z)$ $\displaystyle=\log\prod_{{\mathrel{\mathop{{1\leq i\leq
s}}\limits^{1\leq l\leq
u}}}}\dfrac{\|\tilde{F}(z)\|\cdot\|L^{I}_{il}\|}{|L^{I}_{il}(\tilde{F})(z)|}+\log
c_{4}(z),$

where $c_{3},c_{4}$ are nonzero functions in $\mathcal{C}_{f}$, not depend on
$f$ and $I$, but on $\\{Q_{i}\\}_{i=1}^{q}$. Combining this inequality and
(2.11), we obtain that

(2.12)
$\displaystyle\log\prod_{i=1}^{q}\dfrac{\|\tilde{f}(z)\|^{d}}{|Q_{i}(\tilde{f})(z)|}\leq\frac{\delta}{sa}\left(\max_{I}\log\prod_{{\mathrel{\mathop{{1\leq
i\leq s}}\limits^{1\leq l\leq
u}}}}\dfrac{\|\tilde{F}(z)\|\cdot\|L^{I}_{il}\|}{|L^{I}_{il}(\tilde{F})(z)|}+\log
c_{4}(z)\right)+\log c_{0}(z),$

for all $z$ outside a discrete subset of $\Delta(R)$.

Since $\tilde{F}$ is linearly nondegenerate over ${\mathbb{C}}$, then the
generalized Wronskian

$W(hb_{i}\tilde{\phi}_{J}(\tilde{f}))=\det\left((hb_{i}\tilde{\phi}_{J}(\tilde{f}))^{(k)};1\leq
i\leq t,J\in\mathcal{T}_{L}\right)_{0\leq k\leq tu-1}\not\equiv 0.$

By Theorem 2.4, we have

(2.13) $\displaystyle\begin{split}\Big{\|}_{\gamma}\
\int\limits_{S(r)}\max_{I}\biggl{\\{}\log\prod_{{\mathrel{\mathop{{1\leq i\leq
s}}\limits^{1\leq l\leq
u}}}}\dfrac{\|\tilde{F}(z)\|\cdot\|L^{I}_{il}\|}{|L^{I}_{il}(\tilde{F})(z)|}\biggl{\\}}&\leq
tuT_{F}(r)-N_{W^{\alpha}(hb_{i}\tilde{\phi}_{J}(\tilde{f}))}(r)\\\
&+\dfrac{(tu-1)tu}{2}\Gamma(r)+o(T_{f}(r)),\end{split}$

where $\Gamma(r)=\log\gamma(r)+\epsilon\log r$. Integrating both sides of
(2.12) and using (2.13), we get

(2.14) $\displaystyle\begin{split}\|\
qdT_{f}(r)-\sum_{i=1}^{q}N_{Q_{i}(f)}(r)\leq&\frac{tu\delta}{sa}T_{F}(r)-\frac{\delta}{sa}N_{W^{\alpha}(hb_{i}\tilde{\phi}_{J}(\tilde{f}))}(r)\\\
&+\dfrac{(tu-1)tu}{2}\Gamma(r)+o(T_{f}(r)).\end{split}$

We now estimate the quantity
$\sum_{i=1}^{q}N_{Q_{i}(f)}(r)-\frac{\delta}{sa}N_{W^{\alpha}(hb_{i}\tilde{\phi}_{J}(\tilde{f}))}(r)$.
First, we recall that

$Q_{i}(x)=\sum_{J\in\mathcal{T}_{d}}a_{iJ}x^{I}\in\mathcal{K}[x_{0},\ldots,x_{n}].$

Let $T=(\cdots,t_{kJ},\cdots)\ (k\in\\{1,\ldots,q\\},J\in\mathcal{T}_{d})$ be
a family of variables and

$Q^{T}_{i}=\sum_{J\in\mathcal{T}_{d}}T_{iJ}x^{J}\in\mathbb{Z}[T,x],\ \
i=1,\ldots,q.$

For each element $I_{i}\in\mathcal{I}$, we denote by
$\tilde{R}_{I_{i}}\in\mathbb{Z}[T]$ the resultant of
$\\{Q^{T}_{I_{i}(j)}\\}_{1\leq j\leq l_{i}}$. Then there exist a positive
integer $\lambda$ (common for all $I_{i}$) and polynomials $\tilde{b}_{ijs}\
(0\leq s\leq n,1\leq j\leq l_{i})$ in $\mathbb{Z}[T,x]$, which are zero or
homogeneous in $x$ with degree of $\lambda-d$ such that

$x_{s}^{\lambda}\cdot\tilde{R}_{I_{i}}=\sum_{1\leq j\leq
l_{i}}\tilde{b}_{ijs}Q^{T}_{I_{i}(j)}\ \text{ for all }s\in\\{0,\ldots,n\\},$

and $R_{I_{i}}=\tilde{R}_{I_{i}}(\ldots,a_{kJ},\ldots)\not\equiv 0$. We see
that $R_{I_{i}}\in\mathcal{K}_{f}$ for all $i=1,\ldots,n_{0}$. Set

$b_{ijs}=\tilde{b}_{ijs}((\ldots,a_{jJ},\ldots),(x_{0},\ldots,x_{n})).$

Then we have

$f^{\lambda}_{s}\cdot R_{I_{i}}=\sum_{1\leq j\leq
l_{i}}b_{ijs}(\tilde{f})Q_{I_{i}(j)}(\tilde{f})\ \text{ for all
}i\in\\{0,\ldots,n\\}.$

This implies that

$\nu_{R_{I_{i}}}\geq\min_{1\leq j\leq
l_{i}}\nu_{Q_{I_{i}(j)}(\tilde{f})}+\min_{0\leq s\leq n,1\leq j\leq
l_{i}}\nu_{b_{ijs}(\tilde{f})}.$

We set $R=\prod_{i=1}^{n_{0}}R_{I_{i}}\in\mathcal{K}$. It is easy to see that

$\nu_{b_{ijs}(\tilde{f})}\geq-C\max_{k,J}\nu^{\infty}_{a_{kJ}}$

for a positive constant $C$, and the left hand side of this inequality is only
depend on $\\{Q_{i}\\}$. It implies that there exists a positive constant $c$,
which depends only on $\\{Q_{i}\\}$, such that

$\min_{1\leq j\leq
l_{i}}\nu_{Q_{I_{i}(j)}(f)}\leq\nu_{R}+c\max_{k,J}\nu^{\infty}_{a_{kJ}},$

for all $I_{i}\in\mathcal{I}$.

Fix a point $z_{0}\in\Delta(R)$. Without lose of generality, we may assume
that

$\nu_{Q_{1}(\tilde{f})}(z_{0})\geq\cdots\geq\nu_{Q_{q}(\tilde{f})}(z_{0}),$

and $I_{1}=(1,2,\ldots,q)\in\mathcal{I},l_{1}\leq q-2$. Now, we let
$I=\\{1,\ldots,l_{1}\\}\subset\\{1,\ldots,q\\}$. Then

$\nu_{Q_{j}(f)}(z_{0})\leq\nu_{R}(z_{0})+c\max_{k,J}\nu^{\infty}_{a_{kJ}}(z_{0}),\
j=l_{1},\ldots,q.$

On the other hand, by Lemma 2.7 we have

$\displaystyle\sum_{i=1}^{l_{1}-1}(\nu_{Q_{i}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{Q_{i}(\tilde{f})}(z_{0}))$
$\displaystyle=\sum_{i=1}^{l_{1}}\min\\{0,\nu_{Q_{i}(\tilde{f})}(z_{0})-tu+1\\}$
$\displaystyle\leq\sum_{i=1}^{n}(t_{i+1}-t_{i})\min\\{0,\nu_{Q_{t_{i}}(\tilde{f})}(z_{0})-tu+1\\}$
$\displaystyle\leq\sum_{i=1}^{n}\delta\min\\{0,\nu_{Q_{t_{i}}(\tilde{f})}(z_{0})-tu+1\\}$
$\displaystyle\leq\sum_{i=1}^{n}\delta\min\\{0,\nu_{P_{1,i}(\tilde{f})}(z_{0})-tu+1\\}.$

Therefore,

(2.15)
$\displaystyle\begin{split}\sum_{i=1}^{q}(\nu_{Q_{i}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{Q_{i}(\tilde{f})}(z_{0}))\leq&\sum_{i=1}^{n}\delta\min\\{0,\nu_{P_{1,i}(\tilde{f})}(z_{0})-tu+1\\}\\\
&+(q-N)(\nu_{R}(z_{0})+c\max_{k,J}\nu^{\infty}_{a_{kJ}}(z_{0})).\end{split}$

For $I\in\mathcal{I}$, take linear forms $h_{il}$ in $x^{J}$, $1\leq l\leq
u,s+1\leq i\leq t,J\in\mathcal{T}_{L}$ such that $\\{L^{I}_{il};1\leq l\leq
u,1\leq i\leq s\\}\cup\\{h_{il};1\leq l\leq u,s+1\leq i\leq t\\}$ is linearly
independent over ${\mathbb{C}}$. We easily see that

(2.16)
$\displaystyle\begin{split}\nu_{W^{\alpha}(hb_{i}\tilde{\phi}_{J}(\tilde{f}))}(z_{0})&=\nu_{W^{\alpha}(L^{I}_{il}(\tilde{F}),\ldots,L^{I}_{il}(\tilde{F}))}(z_{0})\\\
&\geq\sum_{{\mathrel{\mathop{{1\leq i\leq s}}\limits^{1\leq l\leq
u}}}}\left(\nu_{L^{I}_{il}(\tilde{F})}(z_{0})-\nu^{[tu-1]}_{L^{I}_{il}(\tilde{F})}(z_{0})\right)\\\
&\geq\sum_{{\mathrel{\mathop{{1\leq i\leq s}}\limits^{1\leq l\leq
u}}}}\left(\nu_{hb_{i}\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{hb_{i}\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0})\right)\\\
&\geq\sum_{{\mathrel{\mathop{{1\leq i\leq s}}\limits^{1\leq l\leq
u}}}}\left(\nu_{\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0})\right)-c_{0}\max_{1\leq
i\leq s}\nu^{\infty}_{hb_{i}}(z_{0}),\end{split}$

where $c_{0}$ is a positive constant, which is chosen independently of $I$,
since there are only finite ordered subset $I$. On the other hand, for each
$1\leq l\leq u,1\leq i\leq s$, since

$\displaystyle\tilde{\psi}^{I}_{l}(\tilde{f})=\frac{1}{c^{I}_{lJ_{l}^{I}}}\prod_{j=1}^{n}P_{i,j}^{i_{jk}}(\tilde{f})h_{l}(\tilde{f}),$

where $(i_{1k},\ldots,i_{nk})=I_{k},h_{l}\in V_{L-d\sigma({\mathbf{i}}_{k})}$,
we have

$\displaystyle\nu_{\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0})\geq\sum_{j=1}^{n}i_{jk}(\nu_{P_{i,j}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{P_{i,j}(\tilde{f})}(z_{0}))-c_{1}\max_{j,J}|\nu_{a_{jJ}}(z_{0})|,$

where $c_{1}$ is a constant and the maximum is taken over all
$a_{jJ}\not\equiv 0$. Summing-up both sides of the above inequalities over all
$1\leq i\leq u,1\leq l\leq s$, we get

(2.17) $\displaystyle\begin{split}\sum_{{\mathrel{\mathop{{1\leq l\leq
s}}\limits^{1\leq i\leq
u}}}}&(\nu_{\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{\tilde{\psi}^{I}_{il}(\tilde{f})}(z_{0}))\\\
&\geq\sum_{j=1}^{n}s\sum_{k=1}^{K}m^{I}_{k}i_{jk}(\nu_{P_{i,j}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{P_{i,j}(\tilde{f})}(z_{0}))-c_{2}\max_{j,J}|\nu_{a_{jJ}}(z_{0})|\\\
&=as\sum_{j=1}^{n}(\nu_{P_{i,j}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{P_{i,j}(\tilde{f})}(z_{0}))-c_{2}\max_{j,J}|\nu_{a_{jJ}}(z_{0})|,\end{split}$

where $c_{2}$ is a constant, which depends only on $\\{Q_{i}\\},t$ and $L$.

Combining (2.16) and (2.17), we get

$\displaystyle\nu_{W^{\alpha}(hb_{i}\tilde{\phi}_{J}(\tilde{f}))}(z_{0})\geq
as\sum_{j=1}^{n}(\nu_{P_{i,j}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{P_{i,j}(\tilde{f})}(z_{0}))-c_{2}\max_{j,J}\nu_{a_{jJ}}(z_{0})-c_{0}\max_{1\leq
i\leq s}\nu^{\infty}_{hb_{i}}(z_{0}).$

Combining (2.16) and this inequality, we obtain

$\displaystyle\frac{\delta}{as}\nu_{W^{\alpha}(b_{i}\tilde{\phi}_{J}(\tilde{f}))}$
$\displaystyle(z_{0})\geq\sum_{i=1}^{q}(\nu_{Q_{i}(\tilde{f})}(z_{0})-\nu^{[tu-1]}_{Q_{i}(\tilde{f})}(z_{0}))$
$\displaystyle-O(\nu_{R}(z_{0})+\max_{j,J}|\nu_{a_{jJ}}(z_{0})|+\max_{1\leq
i\leq s}\nu^{\infty}_{hb_{i}}(z_{0})).$

Integrating both sides of the above inequality, we obtain that

$\|\
\frac{\delta}{as}N_{W^{\alpha}(hb_{i}\tilde{\phi}_{J}(\tilde{f}))}(r)\geq\sum_{i=1}^{q}(N_{Q_{i}(\tilde{f})}(r)-N^{[tu-1]}_{Q_{i}(\tilde{f})}(r))+o(T_{f}(r)).$

From this inequality and (2.14) with a note that $\|\
T_{F}(r)=LT_{f}(r)+o(T_{f}(r))$, we have

(2.18) $\displaystyle\|\
(q-\frac{tuL\delta}{ads})T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N^{[tu-1]}_{Q_{i}(f)}(r)+\frac{(tu-1)tu}{2}\Gamma(r)+o(T_{f}(r)).$

We choose $L:=(n+1)d+2^{n+1}\delta(n+1)^{2}I(\epsilon^{-1})d$ and have some
following estimate:

$\displaystyle a$
$\displaystyle=\sum_{\sigma({\mathbf{i}}_{k})\leq\frac{L}{d}}m^{I}_{k}i_{sk}\geq\sum_{\sigma({\mathbf{i}}_{k})\leq\frac{L}{d}-n}m^{I}_{k}i_{sk}=\dfrac{d^{n}}{n+1}\sum_{\sigma({\mathbf{i}}_{k})\leq\frac{L}{d}-n}\sum_{s=1}^{n+1}i_{sk}$
$\displaystyle=\dfrac{d^{n}}{n+1}\cdot\binom{\frac{L}{d}}{n}\cdot\left(\dfrac{L}{d}-n\right)=d^{n}\binom{\frac{L}{d}}{n+1},$
$\displaystyle\dfrac{uL}{da}$
$\displaystyle\leq\dfrac{\binom{L+n}{n}L}{d^{n+1}\binom{\frac{L}{d}}{n+1}}=(n+1)\prod_{i=1}^{n}\dfrac{L+i}{L-(n-i+1)d}$
$\displaystyle\leq(n+1)\bigl{(}\dfrac{L}{L-(n+1)d}\bigl{)}^{n}=(n+1)\left(1+\dfrac{(n+1)d}{L-(n+1)d}\right)^{n}$
$\displaystyle\leq(n+1)\left(1+\dfrac{2^{n}(n+1)d}{L-(n+1)d}\right)\leq
n+1+\dfrac{\epsilon}{2\delta},$ $\displaystyle\dfrac{tuL}{das}$
$\displaystyle\leq(1+\dfrac{\epsilon}{3(n+1)\delta})(n+1+\dfrac{\epsilon}{2\delta})\leq
n+1+\frac{\epsilon}{\delta}.$

Combining this inequality with (2.18), we get

(2.19) $\displaystyle\|\
\left(q-\delta(n+1)-\epsilon\right)T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N^{[tu-1]}_{Q_{i}(f)}(r)+\frac{(tu-1)tu}{2}\Gamma(r)+o(T_{f}(r)).$

Here we note that:

  * •

$p_{0}:=\left[\dfrac{B-1}{\log(1+\frac{\epsilon}{3(n+1)\delta})}\right]^{2}\leq\left[\dfrac{\binom{L+n}{n}(\binom{L+n}{n}-1)\binom{q}{n}-1}{\log(1+\frac{\epsilon}{3(n+1)\delta})}\right]^{2},$

  * •

$tu-1\leq\binom{L+n}{n}\binom{B+p}{B-1}-1\leq\binom{L+n}{n}p^{B-1}-1\leq\binom{L+n}{n}p_{0}^{\binom{L+n}{n}(\binom{L+n}{n}-1)\binom{q}{n}-2}-1=L_{0}.$

By these estimates and by (2.19), we obtain

$\displaystyle\|\
(q-\delta(n+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N^{[L_{0}]}_{Q_{i}(f)}(r)+\frac{L_{0}(L_{0}+1)}{2}\Gamma(r)+o(T_{f}(r)).$

The theorem is proved. ∎

##  3\. Proof of Theorems 1.2 and 1.3

In order to prove Theorems 1.2 and 1.3 we need the following.

######  Definition 3.1 (see [11]).

Let $V$ be a vector space over a field $\mathcal{K}$. Let $\mathcal{A}$ be a
nonempty subset of $V$. The set $\mathcal{A}$ is said to be non-subdegenerate
if it satisfies

$(\mathcal{A}_{1})_{\mathcal{K}}\cap(\mathcal{A}\setminus\mathcal{A}_{1})_{\mathcal{K}}\neq\\{0\\}\
\forall\ \varnothing\neq\mathcal{A}_{1}\subsetneq\mathcal{A}.$

Here, by the notation $(S)_{\mathcal{K}}$ we denote the linear span over
$\mathcal{K}$ of the subset $S$ of $V$.

######  Lemma 3.2 (see [11]).

Let $\mathcal{A}=\\{v_{1},v_{2},\ldots,v_{q}\\}\ (q\geq 2)$ be a subset of an
$\mathcal{K}$-vector space $V$, which is non-subdegenerate. Then there exist
subsets $I_{1},\ldots,I_{k}$ of $\\{1,\ldots,q\\}$ such that:

$\mathrm{(}i)$ $\\{v_{j}\ |\ j\in I_{1}\\}$ is minimal, and $\\{v_{j}\ |\ j\in
I_{i}\\}$ is linear independent over $\mathcal{K}\ (2\leq i\leq k),$

$\mathrm{(}ii)$
$\left(\\{v_{j}:j\in\bigcup_{i=1}^{k}I_{i}\\}\right)_{\mathcal{K}}=\left(\mathcal{A}\right)_{\mathcal{K}}$
and $I_{i}\cap I_{j}=\emptyset\ (i\neq j)$,

$\mathrm{(}iii)$ For each $2\leq i\leq k,$ there exist meromorphic functions
$c_{i\alpha}\in\mathcal{K}$ $(\alpha\in\bigcup_{j=1}^{i}I_{j})$ such that
$c_{i\alpha}\neq 0$ for all $\alpha\in I_{i}$ and

$\sum_{\alpha\in I_{1}\cup\cdots\cup I_{i}}c_{i\alpha}v_{\alpha}=0.$

Moreover, $n_{1}+n_{2}+\cdots+n_{k}=\mathrm{rank}_{\mathcal{K}}\mathcal{A}-1$,
where $n_{1}=|I_{1}|-2$ and $n_{t}=|I_{t}|-1\ (2\leq t\leq k).$

Here, by the notation $|S|$ we denote the cardinality of the set $S$.

######  Remark 3.3.

(a) We also note that:

$\displaystyle 1+n_{1}$ $\displaystyle=\dim\left(\\{v_{j}:j\in
I_{1}\\}\right)_{\mathcal{K}},$ $\displaystyle 1+n_{1}+n_{2}$
$\displaystyle=\dim\left(\\{v_{j}:j\in
I_{1}\\}\right)_{\mathcal{K}}+\dim\left(\\{v_{j}:j\in
I_{2}\\}\right)_{\mathcal{K}}-1$ $\displaystyle\geq\dim\left(\\{v_{j}:j\in
I_{1}\cup I_{2}\\}\right)_{\mathcal{K}},$ $\displaystyle\cdots$
$\displaystyle\cdots$ $\displaystyle 1+\sum_{i=1}^{k}n_{k}$
$\displaystyle\geq\dim\left(\\{v_{j}:j\in I_{1}\cup\cdots\cup
I_{k}\\}\right)_{\mathcal{K}}=\mathrm{rank}_{\mathcal{K}}\mathcal{A}.$

Then the inequality
$n_{1}+n_{2}+\cdots+n_{k}=\mathrm{rank}_{\mathcal{K}}\mathcal{A}-1$ follows
that

$\dim\left(\left(\\{v_{j}:j\in I_{1}\cup\cdots\cup
I_{s}\\}\right)_{\mathcal{K}}\cap\left(\\{v_{j}:j\in
I_{s+1}\\}\right)_{\mathcal{K}}\right)=1\ (1\leq s\leq k-1).$

(b) In the case where $\mathrm{rank}_{\mathcal{K}}\mathcal{A}\geq 2$: if
$\sharp I_{t}=1$ for some $t>1$ then we remove the subset $I_{t}$ from the
above partition and hence we may suppose that $\sharp I_{t}\geq 2\ \forall
t\geq 2$; if $\sharp I_{1}=2$ then we remove an index from $I_{1}$ and add the
another one into $I_{2}$ to make a new subset $I_{1}$, and hence we may
suppose that $\sharp I_{1}\geq 3.$

Developing the technique of Y. Liu in [4] (also see [19]) to avoid using the
lemma on logarithmic derivative, in [12] we proved the following results,
which is very essential to establish second main theorem with good truncation
level for holomorphic maps on complex discs with finite growth index and
slowly moving hyperplanes.

######  Lemma 3.4 (cf. [12, Lemma 2.3]).

Let $f:\Delta(R)\rightarrow{\mathbb{P}}^{n}({\mathbb{C}})$ be a holomorphic
map, and let $\\{a_{i}\\}_{i=0}^{q-1}$ be $q$ moving hyperplanes in general
position with $\mathrm{rank}_{\mathcal{K}}\\{(\tilde{f},\tilde{a}_{i});0\leq
i\leq q-1\\}=\mathrm{rank}_{\mathcal{K}}(\tilde{f})$, where
$\mathcal{K}=\mathcal{K}_{\\{a_{i}\\}_{i=0}^{q-1}}$. Assume that there exists
a partition $\\{0,\ldots,q-1\\}=I_{1}\cup I_{2}\cup\cdots\cup I_{l}$
satisfying:
$\mathrm{(i)}$ $\sharp I_{1}\geq 3$, $\sharp I_{t}\geq 2\ (2\leq t\leq l)$,
$\\{(\tilde{f},\tilde{a}_{i})\\}_{i\in I_{1}}$ is minimal over $\mathcal{K}$,
$\\{(\tilde{f},\tilde{a}_{i})\\}_{i\in I_{t}}$ is linearly independent over
$\mathcal{R}\ (2\leq t\leq l),$
$\mathrm{(ii)}$ for any $2\leq t\leq l,i\in I_{t},$ there exist meromorphic
functions $c_{i}\in\mathcal{R}\setminus\\{0\\}$ such that

$\sum_{i\in
I_{t}}c_{i}(\tilde{f},\tilde{a}_{i})\in\biggl{(}\bigcup_{j=1}^{t-1}\bigcup_{i\in
I_{j}}(\tilde{f},\tilde{a}_{i})\biggl{)}_{\mathcal{R}}.$

Then we have

$\|_{\gamma}\ T_{f}(r)\leq\sum_{i=1}^{l}\sum_{j\in
I_{i}}N^{[n_{0}]}_{(\tilde{f},a_{j})}(r)+\frac{n_{0}(q-1)}{2}\left((1+\varepsilon)\log\gamma(r)+\varepsilon\log
r\right)+S(r),$

where $S(r)=O(\log T_{f}(r)+\max_{0\leq i\leq q-1}T_{a_{i}}(r))$,
$n_{1}=\sharp I_{1}-2$, $n_{t}=\sharp I_{t}-1$ for $t=2,\ldots,l$ and
$n_{0}=\max_{1\leq i\leq l}n_{i}$.

Here, as usual $(\tilde{f},a)$ and $(\tilde{f},\tilde{a})$ stand for
$a(\tilde{f})$ and $\tilde{a}(\tilde{f})$ respectively (once consider $a$ as a
moving hypersurfaces of degree 1). However, in the proof of [12, Lemma 2.3]
there is a minor gap about computing the multiplicity (the second inequality
in Case 2 of this proof does not hold). Then we need to modify some lines to
correct this proof. For the sake of completeness, we will give the sketch of
the proof to introduce the necessary notations for correcting.

###### Sketch Proof with Correction.

Without loss of generality, we assume that
$I_{i}=\\{t_{i-1}+1,\ldots,t_{i}\\}\ \ (1\leq i\leq l)$, where $t_{0}=-1.$ By
the minimality over $\mathcal{K}$ of the set $\\{(f,\tilde{a}_{i})\\}_{i\in
I_{1}}$, it follows that there exist functions
$c_{i}\in\mathcal{K}\setminus\\{0\\}\ (1\leq i\leq t_{1})$ such that

$(\tilde{f},\tilde{a}_{0})=\sum_{i=1}^{t_{1}}c_{{i}}(\tilde{f},\tilde{a}_{i}).$

We consider the holomorphic map
$F^{1}:\Delta(R)\rightarrow{\mathbb{P}}^{\sharp I_{1}-2}({\mathbb{C}})$ given
by a reduced representation

$F^{1}=\left(\frac{c_{1}}{h_{1}}(\tilde{f},\tilde{a}_{1}),\ldots,\frac{c_{t_{1}}}{h_{1}}(\tilde{f},\tilde{a}_{t_{1}})\right),$

where $h_{1}$ is a nonzero meromorphic function on $\Delta(R)$ with

$\nu_{h_{1}}(z)=\min_{1\leq i\leq t_{1}}\nu_{c_{i}(f,\tilde{a}_{i})}(z)\
\forall z\in\Delta(R).$

For each $i\ (2\leq i\leq l)$, by the assumption (ii), there exists a nonzero
function
$P_{i}\in\biggl{(}\bigcup_{j=1}^{t_{i-1}}(\tilde{f},\tilde{a}_{j})\biggl{)}_{\mathcal{K}}$
such that

$P_{i}=\sum_{j=t_{i-1}+1}^{t_{i}}c_{j}(\tilde{f},\tilde{a}_{j}).$

We consider the holomorphic map $F^{i}:\Delta(R)\to{\mathbb{P}}^{\sharp
I_{i}-1}$ with a reduced representation

$F^{i}=(\dfrac{c_{t_{i-1}+1}}{h_{i}}(\tilde{f},\tilde{a}_{t_{i-1}+1}),\cdots,\dfrac{c_{t_{i}}}{h_{i}}(\tilde{f},\tilde{a}_{t_{i}}))\text{
if }\sharp I_{i}>1,$

and

$F^{i}=(\dfrac{1}{h_{i}}P_{i},\dfrac{c_{t_{i}}}{h_{i}}(\tilde{f},\tilde{a}_{t_{i}}))\text{
if }\sharp I_{i}=1,$

where $h_{i}$ is a meromorphic function on $\Delta(R)$ with

$\nu_{h_{i}}(z)=\min_{t_{i-1}+1\leq j\leq
t_{i}}\nu_{c_{j}(f,\tilde{a}_{j})}(z)\ \forall z\in\Delta(R).$

As [11, Inequality (5)] we have

(3.5) $\displaystyle\begin{split}T_{f}(r)\leq&\sum_{i=1}^{l}\sum_{j\in
I_{i}}N^{[n_{0}]}(r,\nu_{(\tilde{f},a_{j})}-\nu_{i})+N(r,\nu_{1}-\nu)+S(r)\\\
&\ \ +\frac{n_{0}(q-1)}{2}\left((1+\varepsilon)\log\gamma(r)+\varepsilon\log
r\right)-\sum_{i=2}^{l}N(r,\max\\{0,\nu_{P_{i}/h_{i}}-n_{0}\\}),\end{split}$

where $\nu_{i}(z)=\min_{t_{i-1}+1\leq j\leq t_{i}}\nu_{(f,a_{j})}(r)$ and
$\nu(z)=\min_{2\leq j\leq t_{l}}\nu_{(f,a_{j})}(z)$ for all $z\in\Delta(R)$.

Now, for a fixed point $z\in\Delta(R)$ which is neither a pole nor a zero of
any $c_{j},a_{i0}\ (\forall i,j)$, without loss of generality we may assume
that $\left(\frac{c_{1}}{h_{1}}(\tilde{f},\tilde{a}_{1})\right)(z)\neq 0$ and
$\left(\frac{c_{t_{i}+1}(\tilde{f},a_{t_{i}+1})}{h_{i}}\right)(z)\neq 0\
(2\leq i\leq l)$. We distinguish the following two cases:

Case 1: $\nu_{1}(z)\leq n_{0}$. As in the Case 1 of the proof of [11, Lemma
2.3], we have

$\displaystyle\sum_{i=1}^{l}\sum_{j\in
I_{i}}\min\\{n_{0},\nu_{(\tilde{f},a_{j})}(z)-\nu_{i}(z)\\}$
$\displaystyle+\nu_{1}(z)-\sum_{i=2}^{l}\max\\{0,\nu_{P_{i}/h_{i}}(z)-n_{0}\\}$
$\displaystyle\leq\sum_{j=0}^{q-1}\min\\{n_{0},\nu_{(\tilde{f},a_{j})}(z)\\}.$

Case 2: $\nu_{1}(z)>n_{0}$. Let $p$ be the smallest index such that
$\nu_{p}(z)\leq n_{0}$. We have

$\displaystyle\sum_{i=1}^{l}\sum_{j\in
I_{i}}\min\\{n_{0},\nu_{(\tilde{f},a_{j})}(z)-\nu_{i}(z)\\}+\nu_{1}(z)-\sum_{i=2}^{l}\max\\{0,\nu_{P_{i}/h_{i}}(z)-n_{0}\\}$
$\displaystyle\leq\sum_{i=1}^{l}\sum_{j\in
I_{i}}\min\\{n_{0},\nu_{(\tilde{f},a_{j})}(z)\\}-\sum_{i=1}^{p}\min\\{n_{0},\nu_{i}(z)\\}+\nu_{1}(z)$
$\displaystyle\ \ -\sum_{i=2}^{p}\max\\{0,\nu_{i-1}(z)-\nu_{i}(z)-n_{0}\\}$
$\displaystyle\leq\sum_{i=1}^{l}\sum_{j\in
I_{i}}\min\\{n_{0},\nu_{(\tilde{f},a_{j})}(z)\\}-pn_{0}+\nu_{1}(z)-\sum_{i=2}^{p}(\nu_{i-1}(z)-\nu_{i}(z)-n_{0})$
$\displaystyle=\sum_{i=1}^{l}\sum_{j\in
I_{i}}\min\\{n_{0},\nu_{(\tilde{f},a_{j})}(z)\\}-n_{0}+\nu_{p}(z)\leq\sum_{i=1}^{l}\sum_{j\in
I_{i}}\min\\{n_{0},\nu_{(\tilde{f},a_{j})}(z)\\}.$

Hence, we have

$\displaystyle\sum_{i=1}^{l}\sum_{j\in
I_{i}}N^{[n_{0}]}(r,\nu_{(\tilde{f},a_{j})}-\nu_{i})$
$\displaystyle+N(r,\nu_{1}-\nu)+\sum_{i=2}^{l}(N^{[n_{0}]}_{P_{i}/h_{i}}(r)-N_{P_{i}/h_{i}}(r))$
$\displaystyle\leq\sum_{j=0}^{q-1}N^{[n_{0}]}(r,\nu_{(\tilde{f},a_{j})})+S(r).$

This yields that

$\displaystyle
T_{f}(r)\leq\sum_{i=0}^{q-1}N^{[n_{0}]}_{(\tilde{f},a_{i})}(r)+\frac{n_{0}(q-1)}{2}\left((1+\varepsilon)\log\gamma(r)+\varepsilon\log
r\right)+S(r).$

The lemma is proved. ∎

###### Proof of Theorem 1.2.

By replacing $Q_{i}$ with $Q_{i}^{d/d_{i}}$ if necessary, we may suppose that
all $Q_{i}$ are of the same order $d$. Denote by $V^{d}$ the vector space of
all homogeneous polynomials of degree $d$ in $\mathcal{K}[x_{0},\ldots,x_{n}]$
(include the zero polynomial). Set $V^{d}_{f}=\\{Q(\tilde{f})\ ;\ Q\in
V^{d}\\}$, which is a $\mathcal{K}$-vector space. It is seen that $\dim
V^{d}_{f}\leq\dim V^{d}=\binom{n+d}{n}=N+1$.

We denote by $\mathcal{I}$ the set of all permutations of the set
$\\{1,2,\ldots,q\\}$. For each element $I=(i_{1},\ldots,i_{q})\in\mathcal{I}$,
we set

$N_{I}=\\{r\in{\mathbb{R}}^{+};N^{[N]}_{Q_{i_{1}}(\tilde{f})}(r)\leq\cdots\leq
N^{[N]}_{Q_{i_{q}}(\tilde{f})}(r)\\}.$

Consider an element $I=(i_{1},\ldots,i_{q})$ of $\mathcal{I}$, for instance
$I=(1,\ldots,q)$. Set $A=\\{1,\ldots,n(N-n+2)+1\\}$. We cosider a partition
$B_{0},\ldots,B_{l}$ of $A$ satisfying:

(1) $(Q_{j}(\tilde{f}):j\in B_{i})_{\mathcal{K}}\cap(Q_{j}(\tilde{f}):j\in
A\setminus B_{i})_{\mathcal{K}}=\\{0\\}$.

(2) For each $0\leq i\leq l$, $\\{Q_{j}(\tilde{f});j\in B_{i}\\}$ is non-
subdegenerate.

Without loss of generality, we assume that
$|B_{0}|\geq|B_{1}|\geq\cdots\geq|B_{l}|$.

We will show that $|B_{0}|\geq n+1$. Indeed, suppose contrarily that
$|B_{0}|\leq n$. We consider the following two cases.

  * •

Case 1: $d=1$. Then $N=n$ and $|A|=2n+1$. Since $|A\setminus B_{0}|\geq n+1$,
it is clear that

$(Q_{i}(\tilde{f});i\in B_{0})_{\mathcal{K}}\cap(Q_{i}(\tilde{f});i\in
A\setminus B_{0})_{\mathcal{K}}=(Q_{i}(\tilde{f});i\in
B_{0})_{\mathcal{K}}\neq\\{0\\}.$

This is a contradiction.

  * •

Case 2: $d\geq 2$. Consider the function $g(t)=\frac{n(N-n+2)+1}{t}+t$. We
have

$g^{\prime}(t)=1-\frac{n(N-n+2)+1}{t^{2}}\leq\frac{n^{2}-n(N-n+2)-1}{n^{2}}<0\
\forall t\in[1;n]$

(note that $N\geq 2n$). Then $g(t)$ is decreasing on $[1;n]$. Now, for each
$1\leq i\leq l$ we take an index $j_{i}\in B_{i}$. Then
$\\{Q_{j_{1}},\ldots,Q_{j_{l}}\\}\cup\\{Q_{j}:j\in B_{0}\\}$ is linear
independent. Therefore,

$\displaystyle N+1\geq l+|B_{0}|$
$\displaystyle\geq\frac{n(N-n+2)+1}{|B_{0}|}-1+|B_{0}|$
$\displaystyle\geq\frac{n(N-n+2)+1}{n}-1+n=N+1+\frac{1}{n}.$

This is a contradiction.

Then, from the above two cases we must have $|B_{0}|\geq n+1$.

Since $B_{0}$ is non-subdegenerate, we obtain subsets $I_{1},\ldots,I_{k}\
(I_{t}\cap I_{s}=\emptyset,\forall s\neq t)$ of $B_{0}$ with numbers
$n_{1},n_{1},\ldots,n_{k}$ satisfying all conclusions of Lemma 3.4. By Remark
3.3(a), we may assume that $n_{i}\geq 1\ \forall 1\leq i\leq k$. It is clear
that $n_{i}\leq N\ \forall i=1,\ldots,k$.

If $\sharp(I_{1}\cup\cdots\cup I_{k})\leq n+1$, then we set $k_{0}=k$.

If $\sharp(I_{1}\cup\cdots\cup I_{k})>n+1$, we denote by $k_{0}$ the smallest
index such that $\left|I_{1}\cup\cdots\cup I_{k_{0}}\right|\geq n+1$. Then, we
see that $\left\\{Q_{j};j\in I_{1}\cup\cdots\cup I_{k_{0}-1}\right\\}$ is
linear independent. From Remark 3.3(a), we easily see that

$\dim\left((Q_{j};j\in I_{1}\cup\cdots\cup
I_{k_{0}-1})_{\mathcal{K}}\cap(Q_{j};j\in I_{k_{0}})_{\mathcal{K}}\right)\leq
1.$

Therefore

$\sharp(\bigcup_{t=1}^{k_{0}}I_{t})=\dim(Q_{j};j\in I_{1}\cup\cdots\cup
I_{k_{0}-1})_{\mathcal{K}}+\dim(Q_{j};j\in I_{k_{0}})_{\mathcal{K}}=p+1\leq
N+2.$

Consider a holomorphic map
$F:\Delta(R)\rightarrow{\mathbb{P}}^{p+1}({\mathbb{C}})$ with a representation
$\tilde{F}=(Q_{j}(\tilde{f});j\in I_{1}\cup\cdots\cup I_{k_{0}})$. By Lemma
3.4, for every $r\in N_{I}$, we have

$\displaystyle\bigl{\|}\ \ $ $\displaystyle
dT_{f}(r)=T_{F}(r)+o(T_{f}(r))\leq\sum_{i=n(N-n+2)-N}^{n(N-n+2)+1}N^{[N]}_{Q_{j}(f)}(r)+\frac{N(N+1)}{2}\Gamma(r)+o(T_{f}(r))$
$\displaystyle\leq\dfrac{N+2}{q-n(N-n+2)+N+1}\sum_{i=n(N-n+2)-N}^{q}N^{[N]}_{Q_{i}(f)}(r)+\frac{N(N+1)}{2}\Gamma(r)+o(T_{f}(r))$
$\displaystyle\leq\dfrac{N+2}{q-n(N-n+2)+N+1}\sum_{i=1}^{q}N^{[N]}_{Q_{i}(f)}(r)+\frac{N(N+1)}{2}\Gamma(r)+o(T_{f}(r)),$

where $\Gamma(r)=\log\gamma(r)+\epsilon\log r$. We see that
$\bigcup_{i\in\mathcal{I}}N_{I}={\mathbb{R}}^{+}$ and the above inequality
holds for every $r\in N_{I},I\in\mathcal{I}$. This yields that

$\bigl{\|}\ \
T_{f}(r)\leq\dfrac{N+2}{q-n(N-n+2)+N+1}\sum_{i=1}^{q}N^{[N]}_{Q_{i}(f)}(r)+\frac{N(N+1)}{2}\Gamma(r)+o(T_{f}(r)).$

The theorem is proved. ∎

###### Proof of Theorem 1.3.

By repeating the argument as in the proof of Theorem 1.2, it suffices to prove
the theorem for the case where all $Q_{i}$ have the same degree.

Consider arbitrary $(N+2)$ polynomials $Q_{i_{1}},\ldots,Q_{i_{N+2}}\ (1\leq
i_{j}\leq q)$. We see that $n+1\leq\dim(Q_{i_{j}}\ ;\ 1\leq j\leq
N+2)_{\mathcal{K}}\leq N+1<N+2$. Then the set
$\\{Q_{i_{1}},\ldots,Q_{i_{N+2}}\\}$ is of rank at least $n+1$ and is linearly
dependent over $\mathcal{K}$. Hence, there exists a minimal subset over
$\mathcal{K}$, for instance that is $\\{Q_{i_{1}},\ldots,Q_{i_{t}}\\}$, of
$\\{Q_{i_{1}},\ldots,Q_{i_{N+2}}\\}$ with $n+2\leq t\leq N+2$. Then, there
exist nonzero functions $c_{j}\ (1\leq j\leq t)$ in $\mathcal{K}$ such that

$c_{1}Q_{i_{1}}+\cdots+c_{t}Q_{i_{t}}=0.$

Since $Q_{i_{1}},\ldots,Q_{i_{t}}$ are in weakly general position, $t\geq
n+2.$ Denote by $F$ the meromorphic mapping of $\Delta(R)$ into
${\mathbb{P}}^{t-2}({\mathbb{C}})$ which has a representation
$\tilde{F}=(c_{1}Q_{i_{1}}(f),\ldots,c_{t-1}Q_{i_{t-1}}(f))$. Since
$\\{Q_{i_{1}},\ldots,Q_{i_{t}}\\}$ is minimal, $F$ is linearly nondegenerate
over $\mathbb{C}$. Denote by $H_{i}$ the hyperplane in
${\mathbb{P}}^{t-2}({\mathbb{C}})$ defined by the linear from $H_{i}=x_{i}$
$(0\leq i\leq t-2)$, and $H_{t}$ is the hyperplane defined by the linear form
$H_{t}=-x_{0}-\cdots-x_{t-2}$. Applying the second main theorem for these
fixed hyperplanes, we get

$\displaystyle\|\ \ dT_{f}(r)$
$\displaystyle=T_{F}(r)\leq\sum_{j=1}^{t}N^{[t-2]}_{c_{j}Q_{i_{j}}(f)}(r)+\frac{t(t+1)}{2}\Gamma(r)+o(T_{f}(r))$
$\displaystyle\leq\sum_{j=1}^{N+2}N^{[N]}_{Q_{i_{j}}(f)}(r)+\frac{N(N+1)}{2}\Gamma(r)+o(T_{f}(r)),$

where $\Gamma(r)=\log\gamma(r)+\epsilon\log r$. Taking summing-up of both
sides of this inequality over all combinations $\\{i_{1},\ldots,i_{N+2}\\}$
with $1\leq i_{1}<\cdots<i_{N+2}\leq q,$ we have

$\|\ \
dT_{f}(r)\leq\frac{N+2}{q}\sum_{i=1}^{q}N^{[N]}_{Q_{i}(f)}(r)+\frac{N(N+1)}{2}\Gamma(r)+o(T_{f}(r)).$

The theorem is proved. ∎

##  4\. Proof of Theorem 1.4

We need the following result on the Nochka’s weights for families of
hypersurfaces in subgeneral position with respect to a projective varieties,
which is a generalization of the classical results of Nochka [6] (see [8] for
detail).

######  Lemma 4.1 (see [9, Lemma 3.2]).

Let $V$ be a complex projective subvariety of $\mathbb{P}^{n}(\mathbb{C})$ of
dimension $k\ (k\leq n)$. Let $Q_{1},\ldots,Q_{q}$ be $q\ (q>2N-k+1)$
hypersurfaces in $\mathbb{P}^{n}(\mathbb{C})$ in $N$-subgeneral position with
respect to $V$ of the common degree $d.$ Then there are positive rational
constants $\omega_{i}\ (1\leq i\leq q)$ satisfying the following:

i) $0<\omega_{i}\leq 1,\ \forall i\in\\{1,\ldots,q\\}$,

ii) Setting $\tilde{\omega}=\max_{j\in Q}\omega_{j}$, one gets

$\sum_{j=1}^{q}\omega_{j}=\tilde{\omega}(q-2N+k-1)+k+1.$

iii) $\dfrac{k+1}{2N-k+1}\leq\tilde{\omega}\leq\dfrac{k}{N}.$

iv) For $R\subset\\{1,\ldots,q\\}$ with $\sharp R=N+1$, then $\sum_{i\in
R}\omega_{i}\leq k+1$.

v) Let $E_{i}\geq 1\ (1\leq i\leq q)$ be arbitrarily given numbers. For
$R\subset\\{1,\ldots,q\\}$ with $\sharp R=N+1$, there is a subset
$R^{o}\subset R$ such that $\sharp R^{o}=\mathrm{rank}\\{Q_{i}\\}_{i\in
R^{o}}=k+1$ and

$\prod_{i\in R}E_{i}^{\omega_{i}}\leq\prod_{i\in R^{o}}E_{i}.$

###### Proof of Theorem 1.4.

As in the proof of Theorem 1.1, we may suppose that all $Q_{i}\
(i=1,\ldots,q)$ do have the same degree $d$. It is easy to see that there is a
positive constant $\beta$ such that
$\beta\|\tilde{f}\|^{d}\geq|Q_{i}(\tilde{f})|$ for every $1\leq i\leq q.$ Set
$Q:=\\{1,\ldots,q\\}$. Let $\\{\omega_{i}\\}_{i=1}^{q}$ be as in Lemma 4.1 for
the family $\\{Q_{i}\\}_{i=1}^{q}$.

Take a $\mathbb{C}$-basis $\\{[A_{i}]\\}_{i=1}^{H_{V}(d)}$ of $I_{d}(V)$,
where $A_{i}\in H_{d}$. Since $f$ is nondegenerate over $I_{d}(V)$, it implies
that the holomorphic map $F$ with the reduced representation
$\tilde{F}=(A_{1}(\tilde{f}),\ldots,A_{H_{V}(d)}(\tilde{f})\\}$ is linearly
independent over $\mathbb{C}$ and $dT_{f}(r)=T_{F}(r)+O(1)$. Denote by $W$ the
generalized Wronskian of $\tilde{F}$

Let $z$ be a fixed point. There exists $R\subset Q$ with $\sharp R=N+1$ such
that $|Q_{i}(\tilde{f})(z)|\leq|Q_{j}(\tilde{f})(z)|,\forall i\in R,j\not\in
R$. Since $\bigcap_{i\in R}Q_{i}=\varnothing$, there exists a positive
constant $\alpha$ such that

$\alpha\|\tilde{f}\|^{d}(z)\leq\max_{i\in R}|Q_{i}(\tilde{f})(z)|.$

Choose $R^{o}\subset R$ satisfying Lemma 4.1 v) with respect to numbers
$\bigl{\\{}\dfrac{\beta\|\tilde{f}(z)\|^{d}}{|Q_{i}(\tilde{f})(z)|}\bigl{\\}}_{i=1}^{q}$.
Then, we get

$\dfrac{\|\tilde{f}(z)\|^{d(\sum_{i=1}^{q}\omega_{i})}}{|Q_{1}^{\omega_{1}}(\tilde{f})(z)\cdots
Q_{q}^{\omega_{q}}(\tilde{f})(z)|}\leq C\prod_{i\in
R}\left(\dfrac{\beta\|f(z)\|^{d}}{|Q_{i}(\tilde{f})(z)|}\right)^{\omega_{i}}\leq
C^{\prime}\dfrac{\|\tilde{F}\|^{k+1}(z)}{\prod_{i\in
R^{o}}|Q_{i}(\tilde{f})|(z)},$

where $C^{\prime}$ are positive constants (chosen commonly for all $R$).

Therefore, for each $z\in\Delta(R)$, we have

$\displaystyle\log\left(\dfrac{\|f(z)\|^{d(\sum_{i=1}^{q}\omega_{i})}}{|Q_{1}^{\omega_{1}}(\tilde{f})(z)\cdots
Q_{q}^{\omega_{q}}(\tilde{f})(z)|}\right)\leq\log\max_{R^{o}}\dfrac{\|\tilde{F}\|^{k+1}(z)}{\prod_{i\in
R^{o}}|Q_{i}(\tilde{f})|(z)}+O(1),$

where the maximum is taken over all subsets $R^{o}\subset\\{1,\ldots,q\\}$
such that $\\{Q_{i}(\tilde{f});i\in R^{0}\\}$ is linear independent over
${\mathbb{C}}$. Integrating both sides of the above inequality and applying
Theorem 2.1 with the note that
$\sum_{i=1}^{q}\omega_{i}=\tilde{\omega}_{i}(q-2N+k-1)+k+1$, we get

(4.2) $\displaystyle\begin{split}\|_{\gamma}\
&d(\tilde{\omega}(q-2N+k-1)+k+1)T_{f}(r)-\sum_{i=1}^{q}\omega_{i}N_{Q_{i}(\tilde{f})}(r)\\\
&\leq
H_{V}(d)T_{F}(r)-N_{W}(r)+\frac{(H_{V}(d)-1)H_{V}(d)}{2}(\log\gamma(r)+\epsilon\log
r)+o(T_{f}(r)).\end{split}$

As usual argument in Nevanlinna theory, we have

$\sum_{i=1}^{q}\omega_{i}N_{Q_{i}(\tilde{f})}(r)-N_{W}(r)\leq\sum_{i=1}^{q}\omega_{i}N^{[H_{V}(d)-1]}_{Q_{i}(\tilde{f})}(r).$

Combining this inequality and (4.2), we obtain

$\displaystyle\|_{\gamma}\ $ $\displaystyle
d(\tilde{\omega}(q-2N+k-1)+k+1-H_{V}(d))T_{f}(r)$
$\displaystyle\leq\sum_{i=1}^{q}\omega_{i}N^{[H_{V}(d)-1]}_{Q_{i}(\tilde{f})}(r)+\frac{(H_{V}(d)-1)H_{V}(d)}{2}(\log\gamma(r)+\epsilon\log
r)+o(T_{f}(r)).$

Since $\dfrac{k+1}{2N-k+1}\leq\tilde{\omega}\leq\frac{k}{N}$, the above
inequality implies that

$\displaystyle\bigl{\|}_{\gamma}\ $
$\displaystyle\left(q-\frac{(2N-k+1)H_{V}(d)}{k+1}\right)T_{f}(r)$
$\displaystyle\leq\sum_{i=1}^{q}\dfrac{1}{d}N^{[H_{V}(d)-1]}_{Q_{i}(f)}(r)+\frac{(H_{V}(d)-1)H_{V}(d)N}{2dk}(\log\gamma(r)+\epsilon\log
r)+o(T_{f}(r)).$

The theorem is proved. ∎

## 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, Vietnam. 2Thang Long Institute of Mathematics
and Applied Sciences,
Nghiem Xuan Yem, Hoang Mai, HaNoi, Vietnam. E-mail<EMAIL_ADDRESS>