# Meromorphic mappings into projective varieties with arbitrary families of
moving hypersurfaces

Si Duc Quang

###### Abstract.

In this paper, we prove a general second main theorem for meromorphic mappings
into a subvariety $V$ of ${\mathbb{P}}^{N}({\mathbb{C}})$ with an arbitrary
family of moving hypersurfaces. Our second main theorem generalizes and
improves all previous results for meromorphic mappings with moving
hypersurfaces, in particular for meromorphic mappings and families of moving
hypersurfaces in subgeneral position. The method of our proof is different
from that of previous authors used for the case of moving hypersurfaces.

††footnotetext:  2010 Mathematics Subject Classification: Primary 32H30,
32A22; Secondary 30D35.
Key words and phrases: Nevanlinna theory; second main theorem; meromorphic
mapping; hypersurface; homogeneous polynomial; subgeneral position.

##  1\. Introduction

In 1926, R. Nevanlinna [9] initially established a second main theorem for
meromorphic functions and fixed values in the complex plane ${\mathbb{C}}$.
Later on, in 1930 H. Cartan [1] extended the result of Nevanlinna to the case
of linearly nondegenerate meromorphic mappings from ${\mathbb{C}}^{m}$ into
${\mathbb{P}}^{N}({\mathbb{C}})$ with hyperplanes in general position. By
introducing the notion of Nochka’s weights, E. Nochka [10] established the
second main theorem for such mappings with families of hyperplanes in
subgeneral position.

In some few recent years, the second main theorem for hypersurfaces (fixed or
moving) is studied intensively by many authors. Firstly, In 1992, A. E.
Eremenko and M. L. Sodin [5] proved a second main theorem for arbitrary
holomorphic curve from ${\mathbb{C}}$ into ${\mathbb{P}}^{N}({\mathbb{C}})$
intersecting $q$ hypersurfaces $\\{Q_{i}\\}_{i=1}^{q}$ in general position. In
2004 M. Ru [17] considered the case of algebraically nondegenerate meromorphic
mappings from ${\mathbb{C}}^{m}$ into ${\mathbb{P}}^{N}({\mathbb{C}})$ with
families of hypersurfaces in general position by using the filtration of the
vector space of homogeneous polynomial of P. Corvaja and U. Zannier [2]. Then,
in [18] he generalized his results to the case of algebraically nondegenerate
meromorphic mapping into a projective subvariety of
${\mathbb{P}}^{N}({\mathbb{C}})$ by using the method in Diophantine
approximation proposed by J. Evertse and R. Ferretti [6]. His results are
generalized by S. D. Quang for the case of families of hypersurfaces in
subgeneral position in [14], where he proposed the replacing hypersurfaces
method to avoid using Nochka’s weights.

On the other hand, the results of M. Ru also generalized to the case of moving
hypersurfaces by G. Dethloff and T. V. Tan [3, 4]. In oder to establish the
second main theorem for the case of moving hypersurfaces in general position,
G. Dethloff and T. V. Tan have constructed a new filtration instead of the
filtration of Corvaja and Zannier. Also in 2018, S. D. Quang [13] established
the second main theorem for the case of moving hypersurfaces in subgeneral
position. Recently Q. Yan and G. Yu [20] used the methods of [13, 14, 4] to
give a second main theorem for nonconstant meromorphic mappings, but their
second main theorem does not give the normal form of Cartan’s theorem as
usual. Here, we will state here the normal form of the second main theorems
given by Dethloff-Tan and Quang as follows.

We recall the following. Denote by $\mathcal{M}$ the field of all meromorphic
functions on ${\mathbb{C}}^{m}$. We call a moving hypersurface of degree $d$
in ${\mathbb{P}}^{N}({\mathbb{C}})$ a homogeneous polynomial
$Q\in\mathcal{M}[x_{0},\ldots,x_{N}]$ of the form:

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

where $\mathcal{T}_{d}$ is the set of all $n$-tuples $(i_{0},\ldots,i_{N})$
with $i_{0}+\cdots+i_{N}=d$ and $i_{j}\geq 0$ for all $j$,
$a_{I}\in\mathcal{M}\ (I\in\mathcal{T}_{d})$ are not all zeros, ${\bf
x}=(x_{0},\ldots,x_{N})$, ${\bf x}^{I}=x_{0}^{i_{0}}\cdots x_{N}^{i_{N}}$ for
all $I\in\mathcal{T}_{d}$. For each $z$, which is not pole of any $a_{I}$ and
is not a common zero of all $a_{I}\ (I\in\mathcal{T}_{d})$, we denote by
$Q(z)^{*}$ the support of $Q(z)$, i.e.,

$Q(z)^{*}=\\{(x_{0}:\cdots:x_{N})\in{\mathbb{P}}^{N}({\mathbb{C}})|\sum_{I\in\mathcal{T}_{d}}a_{I}(z){\bf
x}^{I}=0\\}.$

We may consider $Q$ as a meromorphic mapping into
${\mathbb{P}}^{\binom{N+d}{N}-1}({\mathbb{C}})$ with a reduced representation
$(\cdots:ha_{I}:\cdots)$ and denote by $T_{Q}(r)$ its characteristic function,
where $h$ is a suitable meromorphic function. The moving hypersurface $Q$ is
said to be slow with respect to a meromorphic mapping $f$ from
${\mathbb{C}}^{m}$ into ${\mathbb{P}}^{N}({\mathbb{C}})$ if
$\|T_{a_{I}}(r)=o(T_{f}(r))\ \forall I\in\mathcal{T}_{d}$. This implies that
$\|T_{Q}(r)=o(T_{f}(r))$. Here the notion “$\|P$” means the assertion $P$
holds for all $r\in(0,+\infty)$ outside a finite Borel measure set.

Let $\\{Q_{i}\\}_{i=1}^{q}$ be a family of moving hypersurfaces in
${\mathbb{P}}^{N}({\mathbb{C}})$, $\deg Q_{i}=d_{i}$. Assume that

$Q_{i}(z)({\bf x})=\sum_{I\in\mathcal{T}_{d_{i}}}a_{iI}(z){\bf x}^{I}.$

Denote by ${\mathcal{K}}_{\\{Q_{i}\\}_{i=1}^{q}}$ the smallest subfield of
$\mathcal{M}$ (the field of all meromorphic functions on ${\mathbb{C}}^{m}$)
which contains ${\mathbb{C}}$ and all $\frac{a_{iI}}{a_{iJ}}$ with
$a_{iJ}\not\equiv 0$.

Let $V$ be a subvariety of ${\mathbb{P}}^{N}({\mathbb{C}})$ of dimension $n$.
We say that $\\{Q_{i}\\}_{i=1}^{q}$ is in weakly general position in $V$ if
there exists $z\in{\mathbb{C}}^{m}$ such that all $a_{iI}\ (1\leq i\leq q,\
I\in\mathcal{I})$ are holomorphic at $z$, and

$V\cap\bigcap_{j=0}^{n}Q_{i_{j}}(z)^{*}=\varnothing.$

Theorem A (see [4]). Let $f$ be a holomorphic curve from ${\mathbb{C}}$ into a
projective subvariety $V$ in ${\mathbb{P}}^{N}(\mathbf{C})$ of dimension $n$
with a reduced representation ${\bf f}=(f_{0},\ldots,f_{N})$. Let
$\\{Q_{i}\\}_{i=1}^{q}$ be a set of slowly (with respect to $f$) moving
hypersurfaces in weakly general position with respect to $V$ and $\deg
Q_{i}=d_{i}\ (1\leq i\leq q).$ Assume that $f$ is algebraically nondegenerate
over $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$. Then for any $\epsilon>0$,

$\|\ (q-n-1-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N_{Q_{i}({\bf
f})}(r).$

In 2018, by using replacing hypersurface method, S. D. Quang [13] considered
the case of moving hypersurfaces in subgeneral position and got the following
second main theorem.

Theorem B (see [13]). Let $f$ be a nonconstant meromorphic map of
$\mathbf{C}^{m}$ into ${\mathbb{P}}^{n}(\mathbf{C})$ with a reduced
representation ${\bf f}=(f_{0},\ldots,f_{n})$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be
a family of slowly (with respect to $f$) moving hypersurfaces in weakly
$l-$subgeneral position with $\deg Q_{i}=d_{i}\ (1\leq i\leq q).$ Assume that
$f$ is algebraically nondegenerate over $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$.
Then for any $\epsilon>0$,

$\|\
(q-(l-n+1)(n+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{[L_{j}]}_{Q_{i}({\bf
f})}(r)+o(T_{f}(r)),$

where $L_{j}=\frac{1}{d_{j}}L_{0}$ and $L_{0}$ is explicitly estimated.

Our purpose in this paper is to generalize all these above mentioned results
to the case of moving hypersurfaces into a projective subvariety $V$ and an
arbitray family of hypersurfaces $\\{Q_{i}\\}_{i=1}^{q}$. In order to do so,
in Section 3, we introduced the notion of “distributive constant” $\Delta_{V}$
(resp. $\Delta_{f}$) of a family of moving hypersurfaces with respect to $V$
(resp. with respect to a meromorphic mapping $f$), and the notion of
“algebraic dimension $n_{f}$” and “algebraic degree $\delta_{f}$” over the
field $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$ of the mapping $f$. Our main
result is stated as follows.

######  Theorem 1.1.

Let $f$ be a nonconstant meromorphic map of $\mathbf{C}^{m}$ into
${\mathbb{P}}^{N}(\mathbf{C})$ with a reduced representation ${\bf
f}=(f_{0},\ldots,f_{N})$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be a family of slowly
(with respect to $f$) moving hypersurfaces in ${\mathbb{P}}^{N}({\mathbb{C}})$
with $\deg Q_{i}=d_{i}\ (1\leq i\leq q).$ Let $n_{f}$ be the algebraic
dimension of $f$ over $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$ and let
$\Delta_{f}$ be the distributive constant of $\\{Q_{i}\\}_{i=1}^{q}$ with
respect to $f$. Suppose that $f$ has finite algebraic degree over
$\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$. Then for any $\epsilon>0$,

$\|\
(q-\Delta_{f}(n_{f}+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N_{Q_{i}({\bf
f})}(r).$

Hence, if $\Delta_{f}=1$, i.e., $\\{Q_{i}\\}_{i=1}^{q}$ is in weakly general
position with respect to $f$, we will get the sharp second main theorem

$\|\
(q-(n_{f}+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N_{Q_{i}({\bf
f})}(r).$

If we assume further that $f$ comes into a subvariety $V$ of
${\mathbb{P}}^{N}({\mathbb{C}})$ of dimension $n$, and $f$ is algebraically
nondegenerate over $\mathcal{K}$, then $n_{f}=\dim V$,
$\Delta_{f}=\Delta_{V}$, the algebraic degree of $f$ over $\mathcal{K}$ is
equal to the degree of $V$. Hence, we have the following corrolary.

######  Corollary 1.2.

Let $f$ be a nonconstant meromorphic map of $\mathbf{C}^{m}$ into a subvariety
$V$ of ${\mathbb{P}}^{N}(\mathbf{C})$ with a reduced representation ${\bf
f}=(f_{0},\ldots,f_{N})$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be a family of slowly
(with respect to $f$) moving hypersurfaces with $\deg Q_{j}=d_{j}\ (1\leq
i\leq q).$ Assume that $f$ is algebraically nondegenerate over
$\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$. Let $\Delta_{V}$ be the distributive
constant of $\\{Q_{i}\\}_{i=1}^{q}$ with respect to $V$. Then, for every
$\epsilon>0$,

$\|\ (q-\Delta_{V}(\dim
V+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N_{Q_{i}({\bf
f})}(r).$

We see that, if $\Delta_{V}=1$ (i.e., $\\{Q_{i}\\}_{i=1}^{q}$ is in weakly
general position with respect to $V$) we will get Theorem A. If
$\\{Q_{i}\\}_{i=1}^{q}$ is in weakly $l$-subgeneral position with respect to
$V$ then $\Delta_{V}\leq(l-n+1)$, and hence we get an extension of Theorem B.

We would also like to emphasize that, our proof is different from the proofs
of the previous authors, such as Dethloff-Tan [3, 4] and Yan-Yu [20]. In fact,
we will develop the method making use of Chow weight of Evertse-Ferretti [6]
and also Ru [17, 18]. Our proof is simpler than that in [3, 4, 20]. Also, from
our proof, it is impossible to give the truncation level for the counting
functions.

##  2\. Notation and Auxialiary results

(a) Some notation from Nevanlina theory.

We use the following usual notations:

$\displaystyle\|z\|$
$\displaystyle:=\big{(}|z_{1}|^{2}+\dots+|z_{m}|^{2}\big{)}^{1/2}\text{ for
}z=(z_{1},\dots,z_{m})\in\mathbb{C}^{m},$ $\displaystyle{\mathbb{B}}^{m}(r)$
$\displaystyle:=\\{z\in\mathbb{C}^{m}:\|z\|<r\\},$ $\displaystyle S(r)$
$\displaystyle:=\\{z\in\mathbb{C}^{m}:\|z\|=r\\}\ (0<r<\infty),$
$\displaystyle v_{m-1}(z)$
$\displaystyle:=\big{(}dd^{c}\|z\|^{2}\big{)}^{m-1},$
$\displaystyle\sigma_{m}(z)$
$\displaystyle:=d^{c}\log\|z\|^{2}\land\big{(}dd^{c}\log\|z\|^{2}\big{)}^{m-1}\text{on}\quad\mathbb{C}^{m}\setminus\\{0\\}.$

For a divisor $\nu$ on a $\mathbb{C}^{m}$ and a positive integer $M$ or
$M=+\infty$, as usual we denote by $N^{[M]}(r,\nu)$ the counting function of
$\nu$ with multiplicities truncated to level $M$.

For a meromorphic function $\varphi$ on ${\mathbb{C}}^{m}$, denote by
$\nu_{\varphi}$ its divisor of zeros and set

$N_{\varphi}(r)=N(r,\nu_{\varphi}),\
N_{\varphi}^{[M]}(r)=N^{[M]}(r,\nu_{\varphi})\ (r_{0}<r<R_{0}).$

For brevity, we will omit the character [M] if $M=+\infty$.

Let $f:\mathbf{C}^{m}\longrightarrow{\mathbb{P}}^{N}(\mathbf{C})$ be a
meromorphic mapping with a reduced representation ${\bf
f}=(f_{0},\ldots,f_{N})$. Set $\|{\bf
f}\|=\big{(}|f_{0}|^{2}+\dots+|f_{N}|^{2}\big{)}^{1/2}$. The characteristic
function of $f$ is defined by

$\displaystyle T_{f}(r)=\int\limits_{S(r)}\log\|{\bf
f}\|\sigma_{m}-\int\limits_{S(1)}\log\|{\bf f}\|\sigma_{m}.$

Let $\varphi$ be a nonzero meromorphic function on $\mathbf{C}^{m}$, which are
occasionally regarded as a meromorphic map into
${\mathbb{P}}^{1}(\mathbf{C})$. The proximity function of $\varphi$ is defined
by

$m(r,\varphi):=\int_{S(r)}\log\max\ (|\varphi|,1)\sigma_{m}.$

The Nevanlinna’s characteristic function of $\varphi$ is defined as follows

$T(r,\varphi):=N_{\frac{1}{\varphi}}(r)+m(r,\varphi).$

Then

$T_{\varphi}(r)=T(r,\varphi)+O(1).$

The function $\varphi$ is said to be small (with respect to $f$) if $\|\
T_{\varphi}(r)=o(T_{f}(r))$.

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

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

where $a_{I}\in\mathcal{M}$ for all $I\in\mathcal{T}_{d}$. The proximity
function of $f$ with respect to $Q$, denoted by $m_{f}(r,Q)$, is defined by

$m_{f}(r,Q)=\int_{S(r)}\log\dfrac{\|{\bf f}\|^{d}}{|Q({\bf
f})|}\sigma_{m}-\int_{S(r_{0})}\log\dfrac{\|{\bf f}\|^{d}}{|Q({\bf
f})|}\sigma_{m}.$

This definition is independent of the choice of the reduced representation of
$f$.

If $Q$ is a slowly moving hypersurface with respect to $f$, then the first
main theorem in Nevanlinna theory for meromorphic mappings and moving
hypersurfaces is stated as follows:

$dT_{f}(r)=m_{f}(r,Q)+N_{Q({\bf f})}(r)+o(T_{f}(r)).$

######  Lemma 2.1 (Lemma on logarithmic derivative, see [11]).

Let $f$ be a nonzero meromorphic function on $\mathbf{C}^{m}.$ Then

$\biggl{|}\biggl{|}\quad
m\biggl{(}r,\dfrac{\mathcal{D}^{\alpha}(f)}{f}\biggl{)}=O(\log^{+}T(r,f))\
(\alpha\in\mathbb{Z}^{m}_{+}).$

Repeating the argument in (Prop. 4.5 [8]), we have the following.

######  Proposition 2.2 (see [8, Prop. 4.5]).

Let $\Phi_{1},...,\Phi_{k}$ be meromorphic functions on $\mathbf{C}^{m}$ such
that $\\{\Phi_{1},...,\Phi_{k}\\}$ are linearly independent over $\mathbf{C}.$
Then there exists an admissible set, which is uniquely chosen in an explicitly
way,

$\\{\alpha_{i}=(\alpha_{i1},...,\alpha_{im})\\}_{i=1}^{k}\subset\mathbb{Z}^{m}_{+}$

with $|\alpha_{i}|=\sum_{j=1}^{m}|\alpha_{ij}|\leq i-1\ (1\leq i\leq k)$ such
that the following are satisfied:

(i)
$\\{{\mathcal{D}}^{\alpha_{i}}\Phi_{1},...,{\mathcal{D}}^{\alpha_{i}}\Phi_{k}\\}_{i=1}^{k}$
is linearly independent over $\mathcal{M},$ i.e.,
$\det{({\mathcal{D}}^{\alpha_{i}}\Phi_{j})}\not\equiv 0,$

(ii)
$\det\bigl{(}{\mathcal{D}}^{\alpha_{i}}(h\Phi_{j})\bigl{)}=h^{k}\cdot\det\bigl{(}{\mathcal{D}}^{\alpha_{i}}\Phi_{j}\bigl{)}$
for any nonzero meromorphic function $h$ on $\mathbf{C}^{m}.$

The determinant $\det\bigl{(}{\mathcal{D}}^{\alpha_{i}}\Phi_{j}\bigl{)}$ is
usually called the general Wronskian of $\\{\Phi_{1},...,\Phi_{k}\\}$.

(b) Chow weights and Hilbert weights.

We recall the notion of Chow weights and Hilbert weights from [18].

Let $X\subset{\mathbb{P}}^{N}({\mathbb{C}})$ be a projective variety of
dimension $n$ and degree $\delta$. The Chow form of $X$ is the unique
polynomial, up to a constant scalar,

$F_{X}(\textbf{u}_{0},\ldots,\textbf{u}_{n})=F_{X}(u_{00},\ldots,u_{0N};\ldots;u_{n0},\ldots,u_{nN})$

in $N+1$ blocks of variables
$\textbf{u}_{i}=(u_{i0},\ldots,u_{iN}),i=0,\ldots,n$ with the following
properties:

  * •

$F_{X}$ is irreducible in $k[u_{00},\ldots,u_{nN}]$;

  * •

$F_{X}$ is homogeneous of degree $\delta$ in each block
$\textbf{u}_{i},i=0,\ldots,n$;

  * •

$F_{X}(\textbf{u}_{0},\ldots,\textbf{u}_{n})=0$ if and only if $X\cap
H_{\textbf{u}_{0}}\cap\cdots\cap H_{\textbf{u}_{n}}\neq\varnothing$, where
$H_{\textbf{u}_{i}},i=0,\ldots,n$, are the hyperplanes given by

$u_{i0}x_{0}+\cdots+u_{iN}x_{N}=0.$

Let ${\bf c}=(c_{0},\ldots,c_{N})$ be a tuple of real numbers and $t$ be an
auxiliary variable. We consider the decomposition

$\displaystyle F_{X}(t^{c_{0}}u_{00},$
$\displaystyle\ldots,t^{c_{N}}u_{0N};\ldots;t^{c_{0}}u_{n0},\ldots,t^{c_{N}}u_{nN})$
$\displaystyle=t^{e_{0}}G_{0}(\textbf{u}_{0},\ldots,\textbf{u}_{N})+\cdots+t^{e_{r}}G_{r}(\textbf{u}_{0},\ldots,\textbf{u}_{N}),$

with
$G_{0},\ldots,G_{r}\in{\mathbb{C}}[u_{00},\ldots,u_{0N};\ldots;u_{n0},\ldots,u_{nN}]$
and $e_{0}>e_{1}>\cdots>e_{r}$. The Chow weight of $X$ with respect to ${\bf
c}$ is defined by

$\displaystyle e_{X}({\bf c}):=e_{0}.$

For each subset $J=\\{j_{0},\ldots,j_{n}\\}$ of $\\{0,\ldots,N\\}$ with
$j_{0}<j_{1}<\cdots<j_{n},$ we define the bracket

$\displaystyle[J]=[J]({\bf u}_{0},\ldots,{\bf
u}_{n}):=\det(u_{ij_{t}}),i,t=0,\ldots,n,$

where $\textbf{u}_{i}=(u_{i0},\ldots,u_{iN})\ (1\leq i\leq n)$ denote the
blocks of $N+1$ variables. Let $J_{1},\ldots,J_{\beta}$ with
$\beta=\binom{N+1}{n+1}$ be all subsets of $\\{0,\ldots,N\\}$ of cardinality
$n+1$.

Then $F_{X}$ can be written as a homogeneous polynomial of degree $\delta$ in
$[J_{1}],\ldots,[J_{\beta}]$. We may see that for
$\textbf{c}=(c_{0},\ldots,c_{N})\in{\mathbb{R}}^{N+1}$ and for any $J$ among
$J_{1},\ldots,J_{\beta}$,

$\displaystyle\begin{split}[J](t^{c_{0}}u_{00},\ldots,t^{c_{N}}u_{0N},&\ldots,t^{c_{0}}u_{n0},\ldots,t^{c_{N}}u_{nN})\\\
&=t\sum_{j\in
J}c_{j}[J](u_{00},\ldots,u_{0N},\ldots,u_{n0},\ldots,u_{nN}).\end{split}$

For $\textbf{a}=(a_{0},\ldots,a_{N})\in\mathbb{Z}^{N+1}$ we write ${\bf
x}^{\bf a}$ for the monomial $x^{a_{0}}_{0}\cdots x^{a_{N}}_{N}$. Denote by
${\mathbb{C}}[x_{0},\ldots,x_{N}]_{u}$ the vector space of homogeneous
polynomials in ${\mathbb{C}}[x_{0},\ldots,x_{N}]$ of degree $u$ (including
$0$). For an ideal $I$ in ${\mathbb{C}}[x_{0},\ldots,x_{N}]$, we put
$I_{u}:={\mathbb{C}}[x_{0},\ldots,x_{N}]_{u}\cap I$. Let $I(X)$ be the prime
ideal in ${\mathbb{C}}[x_{0},\ldots,x_{N}]$ defining $X$. The Hilbert function
$H_{X}$ of $X$ is defined by, for $u=1,2,\ldots,$

$\displaystyle H_{X}(u):=\dim({\mathbb{C}}[x_{0},\ldots,x_{N}]_{u}/I(X)_{u}).$

By the usual theory of Hilbert polynomials,

$\displaystyle H_{X}(u)=\delta\cdot\frac{u^{N}}{N!}+O(u^{N-1}).$

The $u$-th Hilbert weight $S_{X}(u,{\bf c})$ of $X$ with respect to the tuple
${\bf c}=(c_{0},\ldots,c_{N})\in\mathbb{R}^{N+1}$ is defined by

$\displaystyle S_{X}(u,{\bf c}):=\max\left(\sum_{i=1}^{H_{X}(u)}{\bf
a}_{i}\cdot{\bf c}\right),$

where the maximum is taken over all sets of monomials ${\bf x}^{{\bf
a}_{1}},\ldots,{\bf x}^{{\bf a}_{H_{X}(u)}}$ whose residue classes modulo $I$
form a basis of $k[x_{0},\ldots,x_{N}]_{u}/I_{u}.$

The following theorems are due to J. Evertse and R. Ferretti.

######  Theorem 2.3 (see [6, Theorem 4.1]).

Let $X\subset{\mathbb{P}}^{N}({\mathbb{C}})$ be an algebraic variety of
dimension $n$ and degree $\delta$. Let $u>\delta$ be an integer and let ${\bf
c}=(c_{0},\ldots,c_{N})\in\mathbb{R}^{N+1}_{\geqslant 0}$. Then

$\frac{1}{uH_{X}(u)}S_{X}(u,{\bf c})\geq\frac{1}{(n+1)\delta}e_{X}({\bf
c})-\frac{(2n+1)\delta}{u}\cdot\left(\max_{i=0,\ldots,N}c_{i}\right).$

######  Lemma 2.4 (see [7, Lemma 5.1], also [18]).

Let $Y\subset{\mathbb{P}}^{N}({\mathbb{C}})$ be an algebraic variety of
dimension $n$ and degree $\delta$. Let ${\bf c}=(c_{1},\ldots,c_{q})$ be a
tuple of positive reals. Let $\\{i_{0},\ldots,i_{n}\\}$ be a subset of
$\\{1,\ldots,q\\}$ such that

$Y\cap\\{y_{i_{0}}=\cdots=y_{i_{n}}=0\\}=\varnothing.$

Then

$e_{Y}({\bf c})\geq(c_{i_{0}}+\cdots+c_{i_{n}})\delta.$

##  3\. Distributive constant and lemma on replacing moving hypersurfaces

Let $Q_{1},\ldots,Q_{q}$ be $q$ moving hypersurfaces in
${\mathbb{P}}^{N}({\mathbb{C}})$ given by

$Q_{i}(z)({\bf x})=\sum_{I\in\mathcal{T}_{d_{i}}}a_{iI}(z){\bf x}^{I},$

where ${\bf x}=(x_{0},\ldots,x_{N})$, ${\bf x}^{I}=x_{0}^{i_{0}}\cdots
x_{N}^{i_{N}}$ for $I=(i_{0},\ldots,i_{N})$. 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
$h:\mathbf{C}^{m}\setminus A\longrightarrow[0,+\infty]$, which are of the form

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

where $k,l\in\mathbb{N},\ g_{1},....,g_{l+k}\in\mathcal{K}\setminus\\{0\\}$
and $A\subset\mathbf{C}^{m}$, which may depend on $g_{1},....,g_{l+k}$, is an
analytic subset of codimension at least two. Then, for
$h\in\mathcal{C}_{\mathcal{K}}$ we have

$\int\limits_{S(r)}\log h\sigma_{m}=O(\max T_{a_{iI}/a_{iJ}}(r)).$

Then, we see that for every moving hypersurface $Q$ in
$\mathcal{K}[x_{0},\ldots,x_{N}]$ of degree $d$, we have

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

for some $c\in\mathcal{C}_{\mathcal{K}}$.

Let $\\{R_{1},\ldots,R_{p}\\}$ be a family of nonzero homogeneous polynomials
in $\mathcal{K}[x_{0},\ldots,x_{N}]$. For a point $z$, which is neither zero
nor pole of any nonzero coefficients of $R_{i}\ (1\leq i\leq p)$, we set

$n_{z}=\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}.$

######  Lemma 3.1.

$n_{z}$ are constant outside a proper analytic subset of ${\mathbb{C}}^{m}$.

###### Proof.

Take a point $z_{0}$ so that

$n:=n_{z_{0}}=\min_{z}\dim\bigcap_{i=1}^{p}R_{i}(z)^{*},$

where the minimum is taken over all $z$, which is neither zero nor pole of any
nonzero coefficients of $R_{i}\ (1\leq i\leq q)$. Choose
$H_{1},\ldots,H_{n+1}$ be $n+1$ hyperplanes in
${\mathbb{P}}^{N}({\mathbb{C}})$ such that

$\bigcap_{i=1}^{p}R_{i}(z_{0})^{*}\cap\bigcap_{i=1}^{n+1}H_{i}^{*}=\varnothing.$

This implies that

$\bigcap_{i=1}^{p}R_{i}(z)^{*}\cap\bigcap_{i=1}^{n+1}H_{i}^{*}=\varnothing$

for all $z$ outside a proper analytic subset of ${\mathbb{C}}^{m}$. But this
assertion yields that

$\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}\leq n.$

By the definition of $n$, we have

$\min_{z}\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}=n$

for all $z$ outside a proper analytic subset of ${\mathbb{C}}^{m}$. The lemma
is proved. ∎

Let $f$ be a meromorphic mapping from ${\mathbb{C}}^{m}$ into
${\mathbb{P}}^{N}({\mathbb{C}})$ with a reduced representation ${\bf
f}=(f_{0},f_{1},\ldots,f_{N})$. Assume that $Q_{i}(z)({\bf f}(z))\not\equiv 0$
for all $1\leq i\leq q$. Denote by $I(f({\mathbb{C}}^{m}))$ the ideal of all
polynomials $Q$ in $\mathcal{K}[x_{0},\ldots,x_{N}]$ (including zero
polynomial) such that $Q({\bf f})\equiv 0$. By the coherent of the ring of
$\mathcal{K}[x_{0},\ldots,x_{N}]$, there exists a minimal generating set of
$I(f({\mathbb{C}}^{m}))$, denoted again by $\\{R_{1},\ldots,R_{p}\\}$. Then
there is a positive integer $n$ such that

$n=\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}$

for generic points $z\in{\mathbb{C}}^{m}$. Here, once we say an assertion
holds for generic points, it means that assertion holds for every point in
${\mathbb{C}}^{m}$ outside a proper analytic subset.

Now, take another minimal generating set
$R_{1}^{\prime},\ldots,R_{p^{\prime}}^{\prime}$ of $I(f({\mathbb{C}}^{m}))$.
Let $n^{\prime}$ be the integer such that

$n^{\prime}=\dim\bigcap_{i=1}^{p^{\prime}}R^{\prime}_{i}(z)^{*}$

for generic points $z$. We have

$R^{\prime}_{i}=\sum_{j=1}^{p}a_{ij}R_{i},$

where $a_{ij}\in\mathcal{K}[x_{0},\ldots,x_{N}]$. Therefore, for every point
$z\in{\mathbb{C}}^{m}$, which is neither zero nor pole of any nonzero
functions $a_{ij}$ and any nonzero coefficients of $R_{j},R^{\prime}_{i}$, we
have

$\bigcap_{i=1}^{p}R_{i}(z)^{*}\subset\bigcap_{i=1}^{p^{\prime}}R^{\prime}_{i}(z)^{*}.$

Hence $n\leq n^{\prime}$. Similarly $n^{\prime}\leq n$, and hence
$n=n^{\prime}$.

By this reason, we may define the algebraic dimension of $f$ over the field
$\mathcal{K}$ by a constant $n_{f}$ such that

$n_{f}:=\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}$

for a minimal generating set $\\{R_{1},\ldots,R_{p}\\}$ of
$I(f({\mathbb{C}}^{m}))$ and for generic points $z\in{\mathbb{C}}^{m}$.

Also, by the same arguments, we have

$\bigcap_{i=1}^{p}R_{i}(z)^{*}=\bigcap_{i=1}^{p^{\prime}}R^{\prime}_{i}(z)^{*}$

for generic points $z\in{\mathbb{C}}^{m}$. Then we may define the algebraic
degree of the mapping $f$ as follows:

$\delta_{f}=\min\left\\{a\in\mathbb{Z}_{+}|a\geq\deg\bigcap_{i=1}^{p}R_{i}(z)^{*}\
\forall z\in{\mathbb{C}}^{m}\text{ outside a proper analytic subset}\right\\}$

(here, if there is no positive integer $a$ such that
$a\geq\deg\bigcap_{i=1}^{p}R_{i}(z)^{*}$ for generic points $z$ then
$\delta_{f}=+\infty$). Hence if $f$ is considered as an algebraically
nondegenerate (over $\mathcal{K}$) mapping into a subvariety $V$ of
${\mathbb{P}}^{N}({\mathbb{C}})$ then $\delta_{f}=\deg V$.

######  Lemma 3.2.

With the above notation, let $1\leq j_{1}\leq\cdots\leq j_{k}\leq q$. Suppose
that there exists $z_{0}$, which is not pole of any coefficients of all
$R_{i}\ (1\leq i\leq p),Q_{j_{s}}\ (1\leq s\leq k)$, such that
$\bigcap_{i=1}^{p}R_{i}(z_{0})^{*}\cap\bigcap_{s=1}^{k}Q_{j_{s}}(z_{0})^{*}=\varnothing.$
Then we have

$\bigcap_{i=1}^{p}R_{i}(z)^{*}\cap\bigcap_{s=1}^{k}Q_{j_{s}}(z)^{*}=\varnothing$

for generic points $z$, and there exists a function
$c\in\mathcal{C}_{\mathcal{K}}$ such that

$\|{\bf f}(z)\|\leq c(z)\max_{1\leq s\leq k}\\{Q_{j_{s}}(z)({\bf f}(z))\\}.$

Moreover,

$\bigcap_{i=1}^{p^{\prime}}R^{\prime}_{i}(z)^{*}\cap\bigcap_{s=1}^{k}Q_{j_{s}}(z)^{*}=\varnothing.$

for generic points $z\in{\mathbb{C}}^{m}$.

###### Proof.

We use the same arguments as in [4, Lemma 2.3]. It is clear that

$\bigcap_{i=1}^{p}R_{i}(z)^{*}\cap\bigcap_{s=1}^{k}Q_{j_{s}}(z)^{*}=\varnothing$

for generic points $z$. From [3, Lemma 2.2], there exists a function
$c\in\mathcal{C}_{\mathcal{K}}$ such that

$\displaystyle\|{\bf f}(z)\|$ $\displaystyle\leq c(z)\max\\{R_{1}(z)({\bf
f}(z)),\ldots,R_{p}(z)({\bf f}(z)),Q_{j_{1}}(z)({\bf
f}(z)),\ldots,Q_{j_{k}}(z)({\bf f}(z))\\}$ $\displaystyle=c(z)\max_{1\leq
s\leq k}\\{Q_{j_{s}}(z)({\bf f}(z))\\}.$

Also, as above arguments, we have

$\bigcap_{i=1}^{p^{\prime}}R^{\prime}_{i}(z)^{*}\cap\bigcap_{s=1}^{k}Q_{j_{s}}(z)^{*}\subset\bigcap_{i=1}^{p}R_{i}(z)^{*}\cap\bigcap_{s=1}^{k}Q_{j_{s}}(z)^{*}=\varnothing$

for generic points $z\in{\mathbb{C}}^{m}$. ∎

For a subvariety $V$ and an analytic subset $S$ of
${\mathbb{P}}^{N}({\mathbb{C}})$, the codimension of $S$ in $V$ is defined by

$\mathrm{codim}_{V}S=\dim V-\dim(V\cap S).$

######  Definition 3.3.

With the above notation, we define the distributive constant of the family
$\\{Q_{1},\ldots,Q_{q}\\}$ with respect to $f$ by

$\Delta_{f}:={\mathrel{\mathop{{\max}}\limits_{{\Gamma\subset\\{1,\ldots,q\\}}}}}\dfrac{\sharp\Gamma}{\mathrm{codim}_{\bigcap_{i=1}^{p}R_{i}(z)^{*}}\left(\bigcap_{j\in\Gamma}Q_{j}(z)^{*}\right)}$

for generic points $z\in{\mathbb{C}}^{m}$.

Here, we note that $\dim\varnothing=-\infty$. By Lemma 3.1 and the above
arguments, we see that the above definition is well-defined and does not
depend on the choice of the minimal generating set $R_{1},\ldots,R_{p}$ of
$I(f({\mathbb{C}}^{m})).$

######  Definition 3.4.

Let $V$ be a subvariety of ${\mathbb{P}}^{N}({\mathbb{C}})$ of dimension $n$.
Assume that $V$ is not contained in any $Q_{j}(z)^{*}\ (1\leq j\leq q)$
generically. We define the distributive constant of $\\{Q_{1},\ldots,Q_{q}\\}$
with respect to $V$ by

$\Delta_{V}:={\mathrel{\mathop{{\max}}\limits_{{\Gamma\subset\\{1,\ldots,q\\}}}}}\dfrac{\sharp\Gamma}{\mathrm{codim}_{V}\left(\bigcap_{j\in\Gamma}Q_{j}(z)^{*}\right)}$

for generic points $z\in{\mathbb{C}}^{m}$.

######  Remark 3.5.

If the image of $f$ is contained in $V$ and $Q_{j}\not\in
I(f({\mathbb{C}}^{m}))$ for all $j\in\\{1,\ldots,q\\}$ then
$\Delta_{f}\leq(\dim V-n_{f}+1)\Delta_{V}$.

Indeed, since $f({\mathbb{C}}^{m})\subset V$ and $Q_{j}\not\in
I(f({\mathbb{C}}^{m}))$ for all $j\in\\{1,\ldots,q\\}$, then
$\bigcap_{i=1}^{p}R_{i}(z)^{*}\subset V$ and
$\bigcap_{i=1}^{p}R_{i}(z)^{*}\not\subset\bigcap_{j\in\Gamma}Q_{j}(z)^{*}$
generically, and hence it is clear that

$\displaystyle\left(\dim
V-\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}+1\right)\left(\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}-\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}\cap\bigcap_{j\in\Gamma}Q_{j}(z)^{*}\right)$
$\displaystyle\geq\dim
V-\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}+\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}-\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}\cap\bigcap_{j\in\Gamma}Q_{j}(z)^{*}$
$\displaystyle\geq\dim V-\dim\bigcap_{j\in\Gamma}Q_{j}(z)^{*},$

i.e.,

$\displaystyle\left(\dim
V-\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}+1\right)\cdot\mathrm{codim}_{\bigcap_{i=1}^{p}R_{i}(z)^{*}}\left(\bigcap_{j\in\Gamma}Q_{j}(z)^{*}\right)\geq\mathrm{codim}_{V}\left(\bigcap_{j\in\Gamma}Q_{j}(z)^{*}\right)$

for generic points $z$ and all $\Gamma\subset\\{1,\ldots,q\\}.$ This
straightforwardly implies that

$\Delta_{f}\leq(\dim V-n_{f}+1)\Delta_{V}.$

######  Definition 3.6.

With the above definition, the family $\\{Q_{1},\ldots,Q_{q}\\}$ is said to be
in weakly $l-$subgeneral position with respect to $V$ if for every $1\leq
j_{0}<\cdots<j_{l}\leq q,$

$\bigcap_{s=0}^{l}Q_{s}(z)^{*}\cap V=\varnothing$

for generic points $z\in{\mathbb{C}}^{m}$. The family
$\\{Q_{1},\ldots,Q_{q}\\}$ is said to be in weakly $l-$subgeneral position
with respect to $f$ if for every $1\leq j_{0}<\cdots<j_{l}\leq q,$

$\bigcap_{s=0}^{l}Q_{j_{s}}(z)^{*}\cap\bigcap_{i=1}^{p}R_{i}(z)^{*}=\varnothing$

for generic points $z\in{\mathbb{C}}^{m}$. If $l=\dim V$ (resp. $l=n_{f}$)
then we say that $\\{Q_{1},\ldots,Q_{q}\\}$ is in general position with
respect to $V$ (resp. with respect to $f$).

######  Remark 3.7.

(a) If $Q_{1},\ldots,Q_{q}\ (q\geq m+1)$ are in weakly $l-$subgeneral position
with respect to $V$ then we may see that for every subset
$\\{Q_{j_{1}},\ldots,Q_{j_{k}}\\}\ (1\leq k\leq l)$, one has

$\dim\bigcap_{i=1}^{k}Q_{j_{i}}(z)^{*}\leq\min\\{\dim V-1,l-k\\},$

generically, and hence

$\displaystyle\Delta_{V}$ $\displaystyle\leq\max\biggl{\\{}\dfrac{1}{\dim
V-(\dim V-1)},\ldots,\dfrac{l-\dim V+1}{\dim V-(\dim V-1)},\dfrac{l-\dim
V+2}{\dim V-(\dim V-2)},$ $\displaystyle\hskip 160.0pt\ldots,\dfrac{l}{\dim
V-(l-l)}\biggl{\\}}=l-\dim V+1.$

(b) If $Q_{1},\ldots,Q_{q}\ (q\geq l+1)$ are in weakly $l-$subgeneral position
with respect to $f$, then similarly we have

$\Delta_{f}\leq l-\dim\bigcap_{i=1}^{p}R_{i}(z)^{*}+1=l-n_{f}+1\ (\text{for
generic points }z\in{\mathbb{C}}^{m}).$

The following two lemmas play essential role in our proofs, which have been
used in our recent work [15] for fixed hypersufaces.

######  Lemma 3.8.

Let $V$ be a projective subvariety of ${\mathbb{P}}^{N}({\mathbb{C}})$ of
dimension $n$. Let $Q_{0},\ldots,Q_{l}$ be $l$ hypersurfaces in
${\mathbb{P}}^{N}({\mathbb{C}})$ of the same degree $d\geq 1$, such that
$\bigcap_{i=0}^{l}Q_{i}^{*}\cap V=\varnothing$ and

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

where $t_{0},t_{1},\ldots,t_{n}$ integers with $0=t_{0}<t_{1}<\cdots<t_{n}=l$.
Then there exist $n+1$ hypersurfaces $P_{0},\ldots,P_{n}$ in
${\mathbb{P}}^{N}({\mathbb{C}})$ of the forms

$P_{u}=\sum_{j=0}^{t_{u}}c_{uj}Q_{j},\ c_{uj}\in{\mathbb{C}},\ u=0,\ldots,n,$

such that $\left(\bigcap_{u=0}^{n}P_{u}^{*}\right)\cap V=\varnothing.$

###### Proof.

Set $P_{0}=Q_{0}$. We will construct $P_{1},\ldots,P_{n}$ as follows.

Step 1. Firstly, we will construct $P_{1}$. For each irreducible component
$\Gamma$ of dimension $n-1$ of $P_{0}^{*}\cap V$, we put

$V_{1\Gamma}=\\{c=(c_{0},\ldots,c_{t_{1}})\in{\mathbb{C}}^{t_{1}+1}\ ;\
\Gamma\subset Q_{c}^{*},\text{ where }Q_{c}=\sum_{j=0}^{t_{1}}c_{j}Q_{j}\\}.$

Here, $Q_{c}$ may be zero polynomial and its support $Q^{*}_{c}$ is
${\mathbb{P}}^{N}({\mathbb{C}})$. We see that $V_{1\Gamma}$ is a subspace of
${\mathbb{C}}^{t_{1}+1}$. Since
$\dim\left(\bigcap_{j=0}^{t_{1}}Q_{j}^{*}\right)\cap V\leq n-2$, there exists
$i\ (1\leq i\leq t_{1}+1)$ such that $\Gamma\not\subset Q_{i}^{*}$. Then
$V_{1\Gamma}$ is a proper subspace of ${\mathbb{C}}^{t_{1}+1}$. Since the set
of irreducible components of dimension $n-1$ of $P_{0}^{*}\cap V$ is at most
countable,

${\mathbb{C}}^{t_{1}+1}\setminus\bigcup_{\Gamma}V_{1\Gamma}\neq\varnothing.$

Hence, there exists
$(c_{10},c_{11},\ldots,c_{1t_{1}})\in{\mathbb{C}}^{t_{1}+1}$ such that

$\Gamma\not\subset P_{1}^{*}$

for all irreducible components $\Gamma$ of dimension $n-1$ of $P_{0}^{*}\cap
V$, where $P_{1}=\sum_{j=0}^{t_{1}}c_{1j}Q_{j}.$ This implies that
$\dim\left(P_{0}^{*}\cap P_{1}^{*}\right)\cap V\leq n-2.$

Step 2. For each irreducible component $\Gamma^{\prime}$ of dimension $n-2$ of
$\left(P_{0}^{*}\cap P_{1}^{*}\right)\cap V$, put

$V_{2\Gamma^{\prime}}=\\{c=(c_{0},\ldots,c_{t_{2}})\in{\mathbb{C}}^{t_{2}+1}\
;\ \Gamma\subset Q_{c}^{*},\text{ where
}Q_{c}=\sum_{j=0}^{t_{2}}c_{j}Q_{j}\\}.$

Hence, $V_{2\Gamma^{\prime}}$ is a subspace of ${\mathbb{C}}^{t_{2}+1}$. Since
$\dim\left(\bigcap_{i=0}^{t_{2}}Q_{i}^{*}\right)\cap V\leq n-3$, there exists
$i,(0\leq i\leq t_{2})$ such that $\Gamma^{\prime}\not\subset Q_{i}^{*}$.
Then, $V_{2\Gamma^{\prime}}$ is a proper subspace of ${\mathbb{C}}^{t_{2}+1}$.
Since the set of irreducible components of dimension $n-2$ of
$\left(P_{0}^{*}\cap P_{1}^{*}\right)\cap V$ is at most countable,

${\mathbb{C}}^{t_{2}+1}\setminus\bigcup_{\Gamma^{\prime}}V_{2\Gamma^{\prime}}\neq\varnothing.$

Therefore, there exists
$(c_{20},c_{21},\ldots,c_{2t_{2}})\in{\mathbb{C}}^{t_{2}+1}$ such that

$\Gamma^{\prime}\not\subset P_{3}^{*}$

for all irreducible components of dimension $n-2$ of $P_{0}^{*}\cap
P_{1}^{*}\cap V$, where $P_{3}=\sum_{j=0}^{t_{2}}c_{2j}Q_{j}.$ It implies that
$\dim\left(P_{1}^{*}\cap P_{2}^{*}\cap P_{3}^{*}\right)\cap V\leq n-3.$

Repeating again the above steps, after the $n^{\rm th}$-step we get
hypersurfaces $P_{0},\ldots,P_{n}$ satisfying

$\dim\left(\bigcap_{j=0}^{t}P_{j}^{*}\right)\cap V\leq n-t-1\ (0\leq t\leq
n).$

In particular, $\left(\bigcap_{j=0}^{n}P_{j}^{*}\right)\cap V=\varnothing.$ We
complete the proof of the lemma. ∎

######  Lemma 3.9.

Let $t_{0},t_{1},\ldots,t_{n}$ be $n+1$ integers such that
$1=t_{0}<t_{1}<\cdots<t_{n}$, and let
$\Delta={\mathrel{\mathop{{\max}}\limits_{{1\leq s\leq
n}}}}\dfrac{t_{s}-t_{0}}{s}$. Then for every $n$ real numbers
$a_{0},a_{1},\ldots,a_{n-1}$ with $a_{0}\geq a_{1}\geq\cdots\geq a_{n-1}\geq
1$, we have

$a_{0}^{t_{1}-t_{0}}a_{1}^{t_{2}-t_{1}}\cdots
a_{n-1}^{t_{n}-t_{n-1}}\leq(a_{0}a_{1}\cdots a_{n-1})^{\Delta}.$

###### Proof.

Let $s$ be an index, $1\leq s\leq n$ such that
$\Delta=\dfrac{t_{s}-t_{0}}{s}$.

  * •

For every $1\leq k\leq s-1$, we have
$\Delta=\dfrac{t_{s}-t_{0}}{s}\geq\dfrac{t_{s-k}-t_{0}}{s-k}$. This implies
that

$\Delta=\dfrac{t_{s}-t_{0}}{s}\leq\dfrac{t_{s}-t_{0}-(t_{s-k}-t_{0})}{s-(s-k)}=\dfrac{t_{s}-t_{s-k}}{k},\text{
i.e., }t_{s}-t_{s-k}\geq k\Delta.$

  * •

Similarly, for every $1\leq k\leq n-s$, we have
$\Delta=\dfrac{t_{s}-t_{0}}{s}\geq\dfrac{t_{s+k}-t_{0}}{s+k}$. Then

$\Delta=\dfrac{t_{s}-t_{0}}{s}\geq\dfrac{t_{s+k}-t_{0}-(t_{s}-t_{0})}{(s+k)-s}=\dfrac{t_{s+k}-t_{k}}{k},\text{
i.e., }t_{s+k}-t_{s}\leq k\Delta.$

We set $m_{n}=\Delta$ and define

$\displaystyle m_{n-1}$
$\displaystyle=t_{n}-t_{n-1}+\max\\{0,m_{n}-\Delta\\},$ $\displaystyle
m_{n-2}$ $\displaystyle=t_{n-1}-t_{n-2}+\max\\{0,m_{n-1}-\Delta\\},$
$\displaystyle\ldots$ $\displaystyle\ldots$ $\displaystyle m_{0}$
$\displaystyle=t_{1}-t_{0}+\max\\{0,m_{1}-\Delta\\}.$

We see that the set $\\{i|s\leq i\leq n,m_{i}\leq\Delta\\}$ is not empty since
$m_{n}=\Delta$. We set

$u=\min\\{i|s\leq i\leq n,m_{i}\leq\Delta\\}.$

Suppose that $u>s$ then $m_{j}>\Delta\ (\forall s\leq j\leq u-1)$, and hence
we have the following estimate:

$\displaystyle m_{u-1}$ $\displaystyle=t_{u}-t_{u-1}>\Delta,$ $\displaystyle
m_{u-2}$
$\displaystyle=t_{u-1}-t_{u-2}+m_{u-1}-\Delta=t_{u}-t_{u-2}-\Delta>\Delta,$
$\displaystyle m_{u-3}$
$\displaystyle=t_{u-2}-t_{u-3}+m_{u-2}-\Delta=t_{u}-t_{u-3}-2\Delta>\Delta,$
$\displaystyle\ldots$ $\displaystyle\ldots$ $\displaystyle m_{s}$
$\displaystyle=t_{s+1}-t_{s}+m_{s+1}-\Delta=t_{u}-t_{s}-(u-s-1)\Delta>\Delta.$

This implies that

$t_{u}-t_{s}>(u-s)\Delta.$

This is a contradiction. Therefore $u=s$, and hence $m_{s}\leq\Delta$.

We also have the following estimate:

$\displaystyle m_{s-1}$ $\displaystyle=t_{s}-t_{s-1}\geq\Delta,$
$\displaystyle m_{s-2}$
$\displaystyle=t_{s-1}-t_{s-2}+m_{s-1}-\Delta=t_{s}-t_{s-2}-\Delta\geq
2\Delta-\Delta=\Delta,$ $\displaystyle m_{s-3}$
$\displaystyle=t_{s-2}-t_{s-3}+m_{s-2}-\Delta=t_{s}-t_{s-3}-2\Delta\geq
3\Delta-2\Delta=\Delta,$ $\displaystyle\ldots$ $\displaystyle\ldots$
$\displaystyle m_{1}$
$\displaystyle=t_{2}-t_{1}+m_{2}-\Delta=t_{s}-t_{1}-(s-2)\Delta\geq(s-1)\Delta-(s-2)\Delta=\Delta,$
$\displaystyle m_{0}$
$\displaystyle=t_{1}-t_{0}+m_{1}-\Delta=t_{s}-t_{0}-(s-1)\Delta=s\Delta-(s-1)\Delta=\Delta.$

On the other hand, for each $i\in\\{0,\ldots,n-2\\}$ we have

$\dfrac{a_{i}^{m_{i}}a_{i+1}^{\Delta}}{a_{i}^{t_{i+1}-t_{i}}a_{i+1}^{m_{i+1}}}=\dfrac{a_{i}^{\max\\{0,m_{i+1}-\Delta\\}}}{a_{i+1}^{m_{i+1}-\Delta}}\geq\left(\dfrac{a_{i}}{a_{i+1}}\right)^{\max\\{0,m_{i+1}-\Delta\\}}\geq
1,$

i.e.,

$a_{i}^{t_{i+1}-t_{i}}a_{i+1}^{m_{i+1}}\leq a_{i}^{m_{i}}a_{i+1}^{\Delta}.$

Then, we easily have that

$\displaystyle a_{0}^{t_{1}-t_{0}}a_{1}^{t_{2}-t_{1}}\cdots
a_{n-1}^{t_{n}-t_{n-1}}$
$\displaystyle=a_{0}^{t_{1}-t_{0}}a_{1}^{t_{2}-t_{1}}\cdots
a_{n-2}^{t_{n-1}-t_{n-2}}a_{n-1}^{m_{n-1}}$ $\displaystyle\leq
a_{0}^{t_{1}-t_{0}}a_{1}^{t_{2}-t_{1}}\cdots
a_{n-2}^{t_{n-2}-t_{n-3}}a_{n-2}^{m_{n-2}}a_{n-1}^{\Delta}$ $\displaystyle\leq
a_{0}^{t_{1}-t_{0}}a_{1}^{t_{2}-t_{1}}\cdots
a_{n-2}^{t_{n-3}-t_{n-4}}a_{n-3}^{m_{n-3}}a_{n-2}^{\Delta}a_{n-1}^{\Delta}$
$\displaystyle\ldots$ $\displaystyle\leq
a_{0}^{t_{1}-t_{0}}a_{1}^{m_{1}}a_{2}^{\Delta}\cdots a_{n-1}^{\Delta}$
$\displaystyle\leq a_{0}^{m_{0}}a_{1}^{\Delta}a_{2}^{\Delta}\cdots
a_{n-1}^{\Delta}$ $\displaystyle=(a_{0}a_{1}\cdots a_{n-1})^{\Delta}.$

The lemma is proved. ∎

##  4\. General second main theorems for moving hypersurfaces

In oder to prove the main theorem, we will first prove a general form of
second main theorem for moving hyperplanes.

For a subset $\Psi\subset\mathcal{M}$ we denote by $\mathcal{L}(\Psi)$ the
${\mathbb{C}}$-vector space spanned by $\Psi$ over ${\mathbb{C}}$. Assume that
$q:=\sharp\Psi<\infty$, and $1\in\Psi$. Then for a positive integer $p$, we
set
$\Psi(p)=\\{\varphi_{1}\varphi_{2}\cdots\varphi_{p}|\varphi_{j}\in\Psi;j=1,\ldots,p\\}$.
Then

$1\in\Psi(p),\quad\Psi({p})\subset\Psi({p+1}),\quad\sharp\Psi(p)=\binom{p+q-1}{p}=\binom{p+q-1}{q-1}.$

Let $0<\epsilon<1$ be arbitrarily given. Then there exists a smallest integer,
always denoted by $p$ in this section, such that

$\dfrac{\dim\mathcal{L}(\Psi(p+1))}{\dim\mathcal{L}(\Psi(p))}\leq(1+\epsilon).$

Remark: In [19], we have the following estimate:

(4.1)
$\displaystyle\dim\mathcal{L}(\Psi(p+1))\leq\binom{p+q-1}{q-1}\leq\bigl{[}(1+\epsilon)^{[\frac{q}{\log^{2}(1+\epsilon)}]+1}\bigl{]}.$

######  Theorem 4.2.

Let $f$ be a linearly nondegenerate meromorphic mapping of ${\mathbb{C}}^{m}$
in ${\mathbb{P}}^{N}({\mathbb{C}})$ with a reduced representation ${\bf
f}=(f_{0},...,f_{N})$ and let $H_{1},..,H_{q}$ be $q$ arbitrary slowly (with
respect to $f$) moving hyperplanes in ${\mathbb{P}}^{N}({\mathbb{C}})$. Then
for every $\epsilon>0,$

$\biggl{\|}\ \int_{S(r)}\max_{K}\log\left(\prod_{j\in K}\frac{\|{\bf
f}\|}{|H_{j}({\bf f})|}\right)\sigma_{m}\leq(N+1+\epsilon)T_{f}(r),$

where the maximum is taken over all subsets $K\subset\\{1,...,q\\}$ such that
$\\{H_{j}\ ;\ j\in K\\}$ is linearly independent over the field $\mathcal{M}$.

The version of this theorem for fixed hyperplanes is firstly proved by M. Ru
in [16, Theorem 2.3]. For our purpose in this paper, we need this version for
moving hyperplanes.

###### Proof.

By adding more fixed hyperplanes if necessary, we will prove that

$\biggl{\|}\ \int_{S(r)}\max_{K}\log\left(\prod_{j\in K}\frac{\|{\bf
f}\|}{|H_{j}({\bf f})|}\right)\sigma_{m}\leq(N+1+\epsilon)T_{f}(r),$

where the maximum is taken over all subsets $K\subset\\{1,...,q\\}$ such that
$\sharp K=N+1$ and $\\{H_{j}|j\in K\\}$ is linearly independent over the field
$\mathcal{M}$ of all meromorphic functions on ${\mathbb{C}}^{m}$.

Without loss of generality we may assume that

$H_{i}(z)(x_{0},\ldots,x_{N})=\sum_{j=0}^{N}a_{ij}(z)x_{j},$

where all $a_{ij}$ are small functions with respect to $f$ and $a_{i0}=1$. We
put $\Psi=\\{a_{ij}\\}$. By Lemma 3.3 there exists a positive integer $p$ such
that

$\frac{\dim\mathcal{L}(\Psi(p+1))}{\dim\mathcal{L}(\Psi(p))}\leq
1+\dfrac{\epsilon}{N+1}.$

We put

$s=\dim\mathcal{L}(\Psi(p)),\qquad t=\dim\mathcal{L}(\Psi(p)+1)).$

Let $\\{b_{1},\ldots,b_{s}\\}$ be a base of $\mathcal{L}(\Psi(p))$ and
$\\{b_{1},\ldots,b_{t}\\}$ be a base of $\mathcal{L}(\Psi(p+1))$. Then
$\dfrac{t}{s}\leq 1+\epsilon$ and $\\{b_{j}f_{k}(1\leq i\leq t,0\leq k\leq
N)\\}$ is linearly independent over ${\mathbb{C}}$.

Let $K=\\{i_{1},\ldots,i_{N+1}\\}$ be a set such that $\\{H_{j}|j\in K\\}$ is
linearly independent over $\mathcal{M}$, where $1\leq i_{1}<\cdots<i_{N+1}\leq
q$. It is easy to see that $\\{b_{j}H_{i_{v}}({\bf f})(1\leq j\leq s,1\leq
v\leq N+1)\\}$ is linearly independent over ${\mathbb{C}}$. Then we may choose
$\beta^{kl}_{mj}\in{\mathbb{C}}$ such that there is $C_{K}\in
GL((N+1)t;{\mathbb{C}})$ such that

$\displaystyle\det(b_{j}H_{i_{v}}({\bf f})(1\leq j\leq s,1\leq v\leq
N+1),h_{ul}(s+1\leq l\leq t,0\leq u\leq N))$
$\displaystyle=C_{K}\det(b_{j}f_{k}(1\leq j\leq t,0\leq k\leq N)),$

where $h_{ul}=\sum_{1\leq k\leq t,0\leq l^{\prime}\leq
N}\beta^{kl^{\prime}}_{uj}b_{k}f_{l^{\prime}}(s+1\leq j\leq t,0\leq u\leq N)$,
and $C_{K}$ is a nonzero constant.

Let
$\alpha:=(\alpha_{1},\ldots,\alpha_{(N+1)t})\in(\mathbb{Z}_{+}^{m})^{(N+1)t}$
be the admissible set such that

$W\equiv\det\bigl{(}\mathcal{D}^{\alpha_{w}}b_{j}f_{k}(1\leq j\leq t,0\leq
k\leq N)\bigl{)}_{1\leq w\leq(N+1)t}\not\equiv 0.$

Note that $|\alpha_{i}|\leq(N+1)t-1,\forall 1\leq i\leq(N+1)t$. Set

$W_{K}\equiv\det\bigl{(}\mathcal{D}^{\alpha_{w}}b_{j}H_{i_{v}}({\bf
f}),\mathcal{D}^{\alpha_{w}}h_{ul}\bigl{)},$

where $1\leq j\leq t,1\leq v\leq N+1$, $s+1\leq l\leq t,0\leq u\leq N$, and
$1\leq w\leq(N+1)t$. It is easy to see that $W_{K}=C_{K}\cdot W$.

This implies that, for $K=\\{i_{1},\ldots,i_{N+1}\\}$,

$\displaystyle s\log\left(\prod_{j\in K}\frac{\|{\bf f}\|}{|H_{j}({\bf
f})|}\right)$ $\displaystyle\leq(N+1)t\log\|{\bf
f}\|+\log\left(\frac{1}{(\prod_{j\in K}|H_{j}({\bf f})|)^{s}\|{\bf
f}\|^{(N+1)(t-s)}}\right)$ $\displaystyle\leq(N+1)\log\|{\bf
f}\|+\sum_{K}\log^{+}\left(\frac{|W_{K}|}{(\prod_{j\in K}|H_{j}({\bf
f})|)^{s}\|{\bf f}\|^{(N+1)(t-s)}}\right)$
$\displaystyle\quad\quad-\log|W|+O(1).$

Integrating both sides of the above of the above inequality over $S(r)$ and
applying the lemma on logarithmic derivative, we get

$\displaystyle\biggl{\|}\ \int_{S(r)}\max_{K}\log\left(\prod_{j\in
K}\frac{\|{\bf f}\|}{|H_{j}({\bf f})|}\right)\sigma_{m}$
$\displaystyle\leq\dfrac{(N+1)t}{s}T_{f}(r)-\dfrac{1}{s}N_{W}(r)+o(T_{f}(r))$
$\displaystyle\leq(N+1+\epsilon)T_{f}(r).$

The theorem is proved. ∎

###### Proof of Theorem 1.1.

Take $z_{0}$ be a point such that all coefficients of
$R_{1},\ldots,R_{\lambda}$ are holomorphic at $z_{0}$ and

$\Delta_{f}=\max_{\Gamma\subset\\{1,\ldots,q\\}}\dfrac{\sharp\Gamma}{\dim\bigcap_{i=1}^{\lambda}R_{i}(z_{0})^{*}-\dim\bigcap_{i=1}^{\lambda}R_{i}(z_{0})^{*}\cap\bigcap_{j\in\Gamma}Q_{i}(z_{0})^{*}},$

where $\\{R_{1},\ldots,R_{\lambda}\\}$ is a minimal subset of
$\mathcal{K}[x_{0},\ldots,x_{N}]$, which generates $I(f({\mathbb{C}}^{m}))$.
Let $V_{z}=\bigcap_{i=1}^{\lambda}R_{i}(z)^{*}$, which is a projective
subvarieties of ${\mathbb{P}}^{N}({\mathbb{C}})$ of dimension $n_{f}$ for
generic points $z\in{\mathbb{C}}^{m}$. We may suppose that $\dim
V_{z_{0}}=n_{f}$. It is suffice for us to consider the case where
$\Delta_{f}<\dfrac{q}{n_{f}+1}$. Note that $\Delta_{f}\geq 1$, and hence
$q>n_{f}+1$. If there exists $i\in\\{1,\ldots,q\\}$ such that
$\bigcap_{{\mathrel{\mathop{{j=1}}\limits_{{j\neq
i}}}}}^{q}Q_{j}(z_{0})^{*}\cap V_{z_{0}}\neq\varnothing$ then

$\Delta_{f}\geq\dfrac{q-1}{n_{f}}>\dfrac{q}{n_{f}+1}.$

This is a contradiction. Therefore,
$\bigcap_{{\mathrel{\mathop{{j=1}}\limits_{{j\neq
i}}}}}^{q}Q_{j}(z_{0})^{*}\cap V_{z_{0}}=\varnothing$ and hence
$\bigcap_{{\mathrel{\mathop{{j=1}}\limits_{{j\neq i}}}}}^{q}Q_{j}(z)^{*}\cap
V_{z}=\varnothing$ generically, 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 ${\bf x}=(x_{0},\ldots,x_{N})\in{\mathbb{C}}^{N+1}$,
$z\in{\mathbb{C}}^{m}$.

By usual argument, it is suffice for us to prove the theorem only for the case
where all hypersurfaces $Q_{i}\ (1\leq i\leq q)$ are of the same degree $d$.
We denote by $\mathcal{I}$ the set of all bijections from $\\{0,\ldots,q-1\\}$
into $\\{1,\ldots,q\\}$. Denote by $n_{0}$ the cardinality of $\mathcal{I}$,
$n_{0}=q!$, and we write $\mathcal{I}=\\{I_{1},\ldots,I_{n_{0}}\\}$, where
$I_{i}=(I_{i}(0),\ldots,I_{i}(q-1))\in\mathbb{N}^{q}$ and
$I_{1}<I_{2}<\cdots<I_{n_{0}}$ in the lexicographic order.

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

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

Then, $\Delta_{f}\geq\dfrac{t_{i,u}-t_{i,0}}{u}$ for all $1\leq u\leq n_{f}.$
Denote by $P^{\prime}_{i,0},\ldots,P^{\prime}_{i,n_{f}}$ the hypersurfaces
obtained in Lemma 3.8 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 $\\{P_{i,j}\\}_{j=0}^{n_{f}}$ is a family of moving hypersurfaces in
${\mathbb{P}}^{N}({\mathbb{C}})$ with $P_{i,j}(z_{0})=P^{\prime}_{i,j}$. Then
$\bigcap_{j=0}^{n_{f}}P_{i,j}(z_{0})^{*}\cap V_{z_{0}}=\varnothing$, and hence
$\\{P_{i,j}(z)\\}_{j=0}^{n_{f}}$ is in general position with respect to
$V_{z}$ for generic points $z\in{\mathbb{C}}^{m}$. We may choose a positive
constant $B\geq 1$, commonly for all $I_{i}\in\mathcal{I}$, such that

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

for all $0\leq j\leq N$ and for all ${\bf
x}=(x_{0},\ldots,x_{N})\in{\mathbb{C}}^{N+1}$. Denote by $\mathcal{S}$ the set
of all points $z\in{\mathbb{C}}^{m}$ such that
$\bigcap_{j=0}^{n_{f}}P_{i,j}(z)^{*}\cap V_{z}\neq\varnothing$ for some
$I_{i}$. Then $\mathcal{S}$ is a proper analytic subset of ${\mathbb{C}}^{m}$.

By Lemma 3.2, there exists a function $A\in\mathcal{C}_{\mathcal{K}}$, which
is chosen common for all $I_{i}$, such that

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

Fix an element $I_{i}\in\mathcal{I}$. Denote by $S(i)$ the set of all points

$z\in{\mathbb{C}}\setminus\left\\{\bigcup_{i=1}^{q}Q_{i}(z)({\bf
f}(z))^{-1}(\\{0\\})\cup\bigcup_{{\mathrel{\mathop{{0\leq j\leq
n}}\limits_{{I_{i}\in\mathcal{I}}}}}}P_{i,j}(z)({\bf
f}(z))^{-1}(\\{0\\})\right\\}$

such that

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

Therefore, for generic points $z\in S(i)$, By Lemma 3.9 we have

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

where $C\in\mathcal{C}_{\mathcal{K}}$, chosen commonly for all
$I_{i}\in\mathcal{I}$.

For $z\not\in\mathcal{S}$, consider the mapping $\Phi_{z}$ from $V_{z}$ into
${\mathbb{P}}^{l-1}({\mathbb{C}})\ (l=n_{0}(n_{f}+1))$, which maps a point
${\bf x}=(x_{0}:\cdots:x_{N})\in V_{z}$ into the point $\Phi_{z}({\bf
x})\in{\mathbb{P}}^{l-1}({\mathbb{C}})$ given by

$\Phi_{z}({\bf
x})=(P_{1,0}(z)(x):\cdots:P_{1,n_{f}}(z)(x):\cdots:P_{n_{0},0}(z)(x):\cdots:P_{n_{0},n_{f}}(z)(x)),$

where $x=(x_{0},\ldots,x_{N})$. Set

$\tilde{\Phi}_{z}(x)=(P_{1,0}(z)(x),\ldots,P_{1,n_{f}}(z)(x),\ldots,P_{n_{0},0}(z)(x),\ldots,P_{n_{0},n_{f}}(z)(x)).$

Let $Y_{z}=\Phi_{z}(V_{z})$. Since
$V_{z}\cap\bigcap_{j=0}^{n_{f}}P_{1,j}(z)^{*}=\varnothing$, $\Phi_{z}$ is a
finite morphism on $V_{z}$ and $Y_{z}$ is a complex projective subvariety of
${\mathbb{P}}^{l-1}({\mathbb{C}})$ with $\dim Y_{z}=n_{f}$ and of degree

$\delta_{z}:=\deg Y_{z}\leq d^{n_{f}}.\deg V_{z}\leq d^{n_{f}}\delta_{f}\
(\text{for generic points }z).$

For every

${\bf
a}=(a_{1,0},\ldots,a_{1,n_{f}},a_{2,0}\ldots,a_{2,n_{f}},\ldots,a_{n_{0},0},\ldots,a_{n_{0},n_{f}})\in\mathbb{Z}^{l}_{\geq
0}$

and

${\bf
y}=(y_{1,0},\ldots,y_{1,n_{f}},y_{2,0}\ldots,y_{2,n_{f}},\ldots,y_{n_{0},0},\ldots,y_{n_{0},n_{f}})$

we denote ${\bf y}^{\bf a}=y_{1,0}^{a_{1,0}}\ldots
y_{1,n_{f}}^{a_{1,n_{f}}}\ldots y_{n_{0},0}^{a_{n_{0},0}}\ldots
y_{n_{0},n_{f}}^{a_{n_{0},n_{f}}}$. Let $u$ be a positive integer. We set

$\displaystyle\xi_{u}:=\binom{l+u-1}{u}-1,$

and define

$Y_{z,u}:={\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}/(I_{Y_{z}})_{u},$

which is a vector space of dimension $H_{Y_{z}}(u)$.

Denote by $I(Y)_{u}$ the subspace of the $\mathcal{K}$-vector space
$\mathcal{K}[y_{1},\ldots,y_{l}]_{u}$ consisting of all homogeneous
polynomials $P\in\mathcal{K}[y_{1},\ldots,y_{l}]_{u}$ (including the zero
polynomial) such that

$P(z)(\Phi_{z}(f(z)))\equiv 0.$

Let $(\tilde{R}_{1},\ldots,\tilde{R}_{p})$ be an ordered $\mathcal{K}$-basis
of $I(Y)_{u}$. By enlarging the analytic set $\mathcal{S}$ if necessary, we
may assume that all zeros and poles of all nonzero coefficients of
$\tilde{R}_{i}\ (1\leq i\leq p)$ are contained in $\mathcal{S}$, also all
above assertions for generic points $z\in{\mathbb{C}}^{m}$ will hold for all
$z\not\in\mathcal{S}$. Choose $\xi_{u}-p$ nonzero monic homogeneous monomial
$v_{1},\ldots,v_{\xi_{u}-p}$ of degree of $u$ in variables
$y_{1},\ldots,y_{l}$ such that
$\\{\tilde{R}_{1},\ldots,\tilde{R}_{p},v_{1},\ldots,v_{\xi_{u}-p}\\}$ is a
$\mathcal{K}$-basis of $\mathcal{K}[y_{1},\ldots,y_{l}]_{u}$.

######  Claim 4.3.

There is a proper analytic subset of ${\mathbb{C}}^{m}$ such that for all $z$
outside this set, the family of equivalent classes of
$v_{1},\ldots,v_{\xi_{u}-p}$ is a basis of $Y_{z,u}$ and the family
$\\{\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z)\\}$ is a basis of
$(I_{Y_{z}})_{u}$.

Indeed, set $V_{j}(z)(x)=v_{j}(\tilde{\Phi}_{z}(x)),$ which is a homogeneous
polynomial in $\mathcal{K}[x_{0},\ldots,x_{N}]_{ud}$. We see that
$V_{1},\ldots,V_{\xi_{u}-p}$ are linearly independent over $\mathcal{K}$.
There exists $p^{\prime}$ nonzero monic homogeneous monomial
$W_{1},\ldots,W_{p^{\prime}}$ in ${\mathbb{C}}[x_{0},\ldots,x_{N}]_{du}$ such
that $V_{1},\ldots,V_{\xi_{u}-p},W_{1},\ldots,W_{p^{\prime}}$ is an
$\mathcal{K}$-basis of $\mathcal{K}[x_{0},\ldots,x_{N}]_{ud}$. We call
$\\{W_{1},\ldots,W_{p^{\prime}},W_{p^{\prime}+1},\ldots,W_{\xi_{u}-p+p^{\prime}}\\}$
the set of all nonzero monic homogeneous monomials in
${\mathbb{C}}[x_{0},\ldots,x_{N}]_{du}$. Hence, for each
$p^{\prime}<i\leq\xi_{u}-p+p^{\prime}$, we have

$W_{i}=\sum_{j=1}^{\xi_{i}-p}w_{ij}V_{j}+\sum_{j=1}^{p^{\prime}}w^{\prime}_{ij}W_{j},\
(w_{ij},w^{\prime}_{ij}\in\mathcal{K}).$

By enlarging again $S$ if necessary, we may suppose that all zeros and poles
of all nozero coefficients of $w_{ij}$ and $w^{\prime}_{ij}$ belong to
$\mathcal{S}$. Therefore, for every $z\not\in\mathcal{S}$, the family

$\\{V_{1}(z),\ldots,V_{\xi_{u}-p}(z),W_{1},\ldots,W_{p^{\prime}}\\}$

generates the family

$\\{W_{1},\ldots,W_{p^{\prime}},W_{p^{\prime}+1},\ldots,W_{\xi_{u}-p+p^{\prime}}\\},$

and hence it is a basis of ${\mathbb{C}}[x_{0},\ldots,x_{N}]_{du}$. This
implies that $\\{v_{j}(\tilde{\Phi}_{z})\\}_{j=1}^{\xi_{u}-p}$ is
${\mathbb{C}}$-linearly independent, and hence the equivalence classes of
$v_{1},\ldots,v_{\xi_{u}-p}$ modulo $(I_{Y_{z}})_{u}$ are
${\mathbb{C}}$-linear independent in $Y_{z,u}$ for all $z\not\in\mathcal{S}$.

Denote by $\mathcal{T}=\\{T_{1},\ldots,T_{\xi_{u}}\\}$ the set of all nonzero
monic homogeneous monomial of degree of $u$ in variables $y_{1},\ldots,y_{l}$.
Then $\\{T_{1},\ldots,T_{\xi_{u}}\\}$ is an $\mathcal{K}$-basis of
$\mathcal{K}[y_{1},\ldots,y_{l}]_{u}$, and also be an ${\mathbb{C}}$-basis of
${\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}$.

Then for each $1\leq j\leq\xi_{u}$, we have

$T_{j}=\sum_{i=1}^{p}b_{ji}\tilde{R}_{i}+\sum_{i=1}^{\xi_{u}-p}b^{\prime}_{ji}v_{i},$

where $b_{ji},b_{ji}^{\prime}\in\mathcal{K}$. By enlarging $\mathcal{S}$ if
necessary, we may assume that all zeros and poles of all nonzero functions
$b_{ji}$ and $b^{\prime}_{ji}$ are contained in $\mathcal{S}$. Therefore, for
all $z\not\in\mathcal{S}$,
$\\{\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z),v_{1},\ldots,v_{\xi_{u}-p}\\}$
generates the ${\mathbb{C}}$-basis $\\{T_{1},\ldots,T_{\xi_{u}}\\}$ of
${\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}$, and hence it is also
${\mathbb{C}}$-basis of ${\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}$. In particular
$\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z)$ are linearly independent over
${\mathbb{C}}$ for all $z\not\in\mathcal{S}$.

Therefore, for $z\not\in\mathcal{S}$, we have

$\displaystyle\xi_{u}=\dim{\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}$
$\displaystyle=\dim\dfrac{{\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}}{(I_{Y_{z}})_{u}}+\dim(I_{Y_{z}})_{u}$
$\displaystyle\leq\sharp\\{v_{1},\ldots,v_{\xi_{u}-p}\\}+\sharp\\{\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z)\\}=\xi_{u}.$

This yields that
$\dim\dfrac{{\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}}{(I_{Y_{z}})_{u}}=\sharp\\{v_{1},\ldots,v_{\xi_{u}-p}\\}$
and
$\dim(I_{Y_{z}})_{u}=\sharp\\{\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z)\\}$.
Then the set of equivalent classes of $v_{1},\ldots,v_{\xi_{u}-p}$ is a basis
of $Y_{z,u}$, and $\\{\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z)\\}$ are basis
of $(I_{Y_{z}})_{u}$. The claim is proved.

From this claim, we see that

$\xi_{u}-p=H_{Y_{z}}(u)\ \forall z\in{\mathbb{C}}^{m}\setminus\mathcal{S}.$

On the other hand, with the same argument as in the claim, by enlarging
$\mathcal{S}$ if necessary, we may assume that if a subset
$\\{v_{1}^{\prime},\ldots,v^{\prime}_{\xi_{u}-p}\\}$ of $\mathcal{T}$
satisfies
$\\{\tilde{R}_{1},\ldots,\tilde{R}_{p},v_{1}^{\prime},\ldots,v^{\prime}_{\xi_{u}-p}\\}$
is an $\mathcal{K}$-basis of $\mathcal{K}[y_{1},\ldots,y_{l}]_{u}$ then the
set of equivalent classes of $v^{\prime}_{1},\ldots,v^{\prime}_{\xi_{u}-p}$
modulo $(I_{Y_{z}})_{u}$ is a basis of $Y_{z,u}\ \forall z\not\in\mathcal{S}$.

Now, for every subset $\\{v_{1}^{\prime},\ldots,v^{\prime}_{\xi_{u}-p}\\}$ of
$\mathcal{T}$ such that
$\\{\tilde{R}_{1},\ldots,\tilde{R}_{p},v_{1}^{\prime},\ldots,v^{\prime}_{\xi_{u}-p}\\}$
is linearly dependent over $\mathcal{K}$, we take a non-trivial linear
combination

$\sum_{i=1}^{p}c_{i}\tilde{R}_{i}+\sum_{i=1}^{\xi_{u}-p}c^{\prime}_{i}v^{\prime}_{i}\equiv
0,$

where $c_{i},c^{\prime}_{i}\in\mathcal{K}$, not all zeros. By enlarging $S$
again, we may assume that all poles and zeros of $c_{i},c^{\prime}_{i}$ belong
to $\mathcal{S}$. Hence, for all $z\not\in\mathcal{S}$,
$\\{\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z),v_{1}^{\prime},\ldots,v^{\prime}_{\xi_{u}-p}\\}$
is linearly dependent over ${\mathbb{C}}$. Consequently, the set of equivalent
classes of $v_{1}^{\prime},\ldots,v^{\prime}_{\xi_{u}-p}$ modulo
$(I_{Y_{z}})_{u}$ is not a basis of $Y_{z,u}$.

Now, consider the meromorphic mapping $F$ into
${\mathbb{P}}^{\xi_{u}-p}({\mathbb{C}})$ with the representation

${\bf F}=(v_{0}(\tilde{\Phi}\circ{\bf
f}),\ldots,v_{\xi_{u}-p}(\tilde{\Phi}\circ{\bf f})).$

Hence $F$ is linearly nondegenerate over $\mathcal{K}$, since $f$ is
algebraically nondegenerate over $\mathcal{K}$.

Now, we fix an index $i\in\\{1,\ldots,n_{0}\\}$ and a point $z\in
S(i)\setminus\mathcal{S}$. We define

${\bf
c}_{z}=(c_{1,0,z},\ldots,c_{1,n_{f},z},c_{2,0,z},\ldots,c_{2,n_{f},z},\ldots,c_{n_{0},0,z},\ldots,c_{n_{0},n_{f},z})\in\mathbb{Z}^{l},$

where

(4.4) $\displaystyle c_{i,j,z}:=\log\frac{\|{\bf
f}(z)\|^{d}\|P_{i,j}(z)\|}{|P_{i,j}(z)({\bf f}(z))|}\text{ for
}i=1,\ldots,n_{0}\text{ and }j=0,\ldots,n_{f}.$

We see that $c_{i,j,z}\geq 0$ for all $i$ and $j$. By the definition of the
Hilbert weight, there are ${\bf a}_{1,z},\ldots,{\bf
a}_{\xi_{u}-p,z}\in\mathbb{N}^{l}$ with

${\bf
a}_{i,z}=(a_{i,1,0,z},\ldots,a_{i,1,n_{f},z},\ldots,a_{i,n_{0},0,z},\ldots,a_{i,n_{0},n_{f},z}),$

where $a_{i,j,s,z}\in\\{1,\ldots,\xi_{u}\\},$ such that the residue classes
modulo $(I_{Y})_{u}$ of ${\bf y}^{{\bf a}_{1,z}},\ldots,{\bf y}^{{\bf
a}_{\xi_{u}-p,z}}$ form a basic of
${\mathbb{C}}[y_{1},\ldots,y_{l}]_{u}/(I_{Y_{z}})_{u}$ and

(4.5) $\displaystyle S_{Y}(u,{\bf c}_{z})=\sum_{i=1}^{\xi_{u}-p}{\bf
a}_{i,z}\cdot{\bf c}_{z}.$

We see that ${\bf y}^{{\bf a}_{i,z}}\in\mathcal{T}$. Then as above
$\\{\tilde{R}_{1}(z),\ldots,\tilde{R}_{p}(z),{\bf y}^{{\bf
a}_{1,z}},\ldots,{\bf y}^{{\bf a}_{\xi_{u}-p,z}}\\}$ is a basis of
$\mathcal{K}[y_{1},\ldots,y_{l}]$. Therefore, $\\{{\bf y}^{{\bf
a}_{1,z}},\ldots,{\bf y}^{{\bf a}_{\xi_{u}-p,z}}\\}$ is a basis of
$\dfrac{\mathcal{K}[y_{1},\ldots,y_{l}]_{u}}{I(Y)_{u}}$. Then, we may write

${\bf y}^{{\bf a}_{i,z}}=L_{i,z}(v_{0},\ldots,v_{H_{Y}(u)})\ \ \ \text{modulo
}I(Y)_{u},$

where $L_{i,z}\ (1\leq i\leq\xi_{u}-p)$ are independent linear forms with
coefficients in $\mathcal{K}$. We have

$\displaystyle\log\prod_{i=1}^{\xi_{u}-p}|L_{i,z}({\bf F}(z))|$
$\displaystyle=\log\prod_{i=1}^{\xi_{u}-p}\prod_{{\mathrel{\mathop{{0\leq
j\leq n_{f}}}\limits^{1\leq t\leq n_{0}}}}}|P_{t,j}({\bf
f}(z))|^{a_{i,t,j,z}}$ $\displaystyle=-S_{Y}(u,{\bf
c}_{z})+du(\xi_{u}-p)\log\|{\bf f}(z)\|+O(u(\xi_{u}-p)).$

It follows that

$\displaystyle\log\prod_{i=1}^{\xi_{u}-p}\dfrac{\|{\bf
F}(z)\|\cdot\|L_{i,z}\|}{|L_{i,z}({\bf F}(z))|}=$ $\displaystyle S_{Y}(u,{\bf
c}_{z})-du(\xi_{u}-p)\log\|{\bf f}(z)\|$ $\displaystyle+(\xi_{u}-p)\log\|{\bf
F}(z)\|+O(u(\xi_{u}-p)).$

Note that the number of these linear forms $L_{i,z}$ is finite. Denote by
$\mathcal{L}$ the set of all $L_{i,z}$ occurring in the above inequalities. We
have

(4.6) $\displaystyle\begin{split}S_{Y}(u,{\bf
c}_{z})\leq&\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf f}(z))|}+du(\xi_{u}-p)\log\|{\bf f}(z)\|\\\
&-(\xi_{u}-p)\log\|{\bf F}(z)\|+O(u(\xi_{u}-p)),\end{split}$

where the maximum is taken over all subsets $\mathcal{J}\subset\mathcal{L}$
with $\sharp\mathcal{J}=H_{Y}(u)$ and $\\{L|L\in\mathcal{J}\\}$ is linearly
independent over $\mathcal{K}$. From Theorem 2.3 we have

(4.7) $\displaystyle\dfrac{1}{u(\xi_{u}-p)}S_{Y_{z}}(u,{\bf c}_{z})\geq$
$\displaystyle\frac{1}{(n_{f}+1)\delta_{z}}e_{Y_{z}}({\bf
c}_{z})-\frac{(2n_{f}+1)\delta_{z}}{u}\max_{{\mathrel{\mathop{{1\leq i\leq
n_{0}}}\limits_{{0\leq j\leq n_{f}}}}}}c_{i,j,z}$

We chose an index $i_{0}$ such that $z\in S(i_{0})$. It is clear that

$\displaystyle\max_{{\mathrel{\mathop{{1\leq i\leq n_{0}}}\limits_{{0\leq
j\leq n_{f}}}}}}c_{i,j,z}\leq\sum_{0\leq j\leq n_{f}}\log\frac{\|{\bf
f}(z)\|^{d}\|P_{i_{0},j}\|}{|P_{i_{0},j}(z)({\bf f}(z))|}+O(1),$

where the term $O(1)$ does not depend on $z$ and $i_{0}$. Combining (4.6),
(4.7) and the above remark, we get

$\displaystyle\frac{1}{(n_{f}+1)\delta_{z}}e_{Y_{z}}({\bf c}_{z})\leq$
$\displaystyle\dfrac{1}{u(\xi_{u}-p)}\left(\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}-(\xi_{u}-p)\log\|{\bf F}(z)\|\right)$
(4.8) $\displaystyle\begin{split}&+d\log\|{\bf
F}(z)\|+\frac{(2n_{f}+1)\delta_{z}}{u}\max_{{\mathrel{\mathop{{1\leq i\leq
n_{0}}}\limits_{{0\leq j\leq n_{f}}}}}}c_{i,j,z}+O(1/u)\\\
\leq&\dfrac{1}{u(\xi_{u}-p)}\left(\max_{\mathcal{J}\subset\mathcal{L}}\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}-(\xi_{u}-p)\log\|{\bf F}(z)\|\right)\\\
&+d\log\|{\bf F}(z)\|+\frac{(2n_{f}+1)\delta_{z}}{u}\sum_{0\leq j\leq
n_{f}}\log\frac{\|{\bf f}(z)\|^{d}\|P_{i_{0},j}\|}{|P_{i_{0},j}(z)({\bf
f}(z))|}+O(1/u).\end{split}$

Since $\\{P_{i_{0},0}(z)=\cdots=P_{i_{0},n_{f}}(z)=0\\}\cap
V_{z}=\varnothing$, by Lemma 2.4, we have

(4.9) $\displaystyle e_{Y_{z}}({\bf
c}_{z})\geq(c_{i_{0},0,z}+\cdots+c_{i_{0},n_{f},z})\cdot\delta_{z}=\left(\sum_{0\leq
j\leq n_{f}}\log\frac{\|{\bf
f}(z)\|^{d}\|P_{i_{0},j}(z)\|}{|P_{i_{0},j}(z)({\bf
f}(z))|}\right)\cdot\delta_{z},$

for all $z\not\in\mathcal{S}$.

Then, from (4.4), (4.8) and (4.9) we have

$\displaystyle\frac{1}{\Delta_{f}}\log\prod_{i=1}^{q}\dfrac{\|{\bf
f}(z)\|^{d}}{|Q_{i}(z)({\bf f}(z))|}$
$\displaystyle\leq\dfrac{n_{f}+1}{u(\xi_{u}-p)}\left(\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}-(\xi_{u}-p)\log\|{\bf F}(z)\|\right)$
$\displaystyle+\frac{(2n_{f}+1)(n_{f}+1)\delta_{z}}{u}\sum_{{\mathrel{\mathop{{1\leq
i\leq n_{0}}}\limits_{{0\leq j\leq n_{f}}}}}}\log\frac{\|{\bf
f}(z)\|^{d}\|P_{i,j}(z)\|}{|P_{i,j}(z)({\bf f}(z))|}$
$\displaystyle+d(n_{f}+1)\log\|{\bf f}(z)\|+\frac{1}{\Delta_{f}}\log
C(z)+O(1)$

for generic points $z\in{\mathbb{C}}^{m}$, where the term $O(1)$ does not
depend on $z$. Integrating both sides of the above inequality, we obtain

$\displaystyle\frac{1}{d}\sum_{i=1}^{q}m_{f}(r,Q_{i})\leq$
$\displaystyle\dfrac{\Delta_{f}(n_{f}+1)}{du(\xi_{u}-p)}\left(\int\limits_{S(r)}\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}\sigma_{m}-(\xi_{u}-p)T_{F}(r)\right)$
(4.10)
$\displaystyle+\Delta_{f}(n_{f}+1)T_{f}(r)+\frac{\Delta_{f}(2n_{f}+1)(n_{f}+1)\delta_{f}}{du}\sum_{{{\mathrel{\mathop{{1\leq
i\leq n_{0}}}\limits_{{0\leq j\leq n_{f}}}}}}}m_{f}(r,P_{i,j})$

(note that $\delta_{z}\leq d^{n_{f}}\delta_{f}$ for all
$z\not\in\mathcal{S}$). On the other hand, by Theorem 4.2, we have

(4.11) $\displaystyle\begin{split}\biggl{\|}\
\int\limits_{S(r)}\max_{\mathcal{J}\subset\mathcal{L}}&\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf F}(z))|}\sigma_{m}-(\xi_{u}-p)T_{F}(r)\\\
&\leq\dfrac{\epsilon(\xi_{u}-p)}{2\Delta_{f}(n_{f}+1)}T_{F}(r)\leq\dfrac{du\epsilon(\xi_{u}-p)}{2\Delta_{f}(n_{f}+1)}T_{f}(r).\end{split}$

Combining (4.10) with (4.11), we have

$\displaystyle\bigl{\|}\ (q-\Delta_{f}(n_{f}+1)\bigl{)}T_{f}(r)$
$\displaystyle\leq\sum_{i=1}^{q}\frac{1}{d}N_{Q_{i}({\bf
f})}(r)+\frac{\Delta_{f}(2n_{f}+1)(n_{f}+1)\delta_{f}}{ud}\sum_{{{\mathrel{\mathop{{1\leq
i\leq n_{0}}}\limits_{{0\leq j\leq n_{f}}}}}}}m_{f}(r,P_{i,j})$
$\displaystyle\ \ \ +\dfrac{\epsilon}{2}T_{f}(r)$
$\displaystyle\leq\sum_{i=1}^{q}\frac{1}{d}N_{Q_{i}({\bf f})}(r)+\epsilon
T_{f}(r)\ \ (\text{ by choosing $u$ large enough}).$

The theorem is proved. ∎

######  Remark 4.12.

(a) By the proof of Theorem 4.2, the inequality (4.11) can be replaced by

$\biggl{\|}\
\int\limits_{S(r)}\max_{\mathcal{J}\subset\mathcal{L}}\log\prod_{L\in\mathcal{J}}\dfrac{\|{\bf
F}(z)\|\cdot\|L\|}{|L({\bf
F}(z))|}\sigma_{m}-(\xi_{u}-p)T_{F}(r)\leq\dfrac{1}{s}N_{W(F)}(r)+\dfrac{\epsilon(\xi_{u}-p)}{2\Delta_{f}(n_{f}+1)}T_{F}(r).$

where $W(F)$ is a general Wronskian of $\\{b_{j}v_{i}(\tilde{\Phi}\circ{\bf
f});1\leq j\leq t,0\leq i\leq n_{u}\\}$; $\\{b_{1},\ldots,b_{t}\\}$ is a basis
of $\Psi(q^{\prime}+1)$ with a positive integer $q^{\prime}$, $\Psi$ is the
set of all coefficients of $Q_{i}\ (1\leq i\leq q)$ and
$s=\dim\mathcal{L}(\Psi(q^{\prime}))$. Then, the second main theorem can be
obtained as follows:

$\|\
(q-\Delta_{f}(n_{f}+1)-\epsilon\bigl{)}T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N_{Q_{i}({\bf
f})}(r)-\dfrac{1}{s}N_{W(F)}(r).$

Since the number $q^{\prime}$ and the dimensions
$s=\dim\mathcal{L}(\Psi(q^{\prime})),t=\dim\mathcal{L}(\Psi(q^{\prime}+1))$
can be explicitly estimated as (4.1) (bound above by an function of $q$ and
$\epsilon$), the right hand side of the above inequality will be bounded above
by a sum of the form $\sum_{i=1}^{q}\frac{1}{d}N^{[L(q,\epsilon)]}_{Q_{i}({\bf
f})}(r)$, where $L(q,\epsilon)$ is explicitly estimated. Hence, we may get the
second main theorem with truncation level for counting functions as follows.

######  Theorem 4.13.

Let $f$ be a nonconstant meromorphic map of $\mathbf{C}^{m}$ into
${\mathbb{P}}^{N}(\mathbf{C})$ with a reduced representation ${\bf
f}=(f_{0},\ldots,f_{N})$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be a family of slowly
(with respect to $f$) moving hypersurfaces in ${\mathbb{P}}^{N}({\mathbb{C}})$
with $\deg Q_{i}=d_{i}\ (1\leq i\leq q).$ Let $n_{f}$ be the algebraic
dimension of $f$ over $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$ and let
$\Delta_{f}$ be the distributive constant of $\\{Q_{i}\\}_{i=1}^{q}$ with
respect to $f$. Suppose that $f$ has finite algebraic degree over
$\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$. Then for any $\epsilon>0$, there exists
a positive integer $L(q,\epsilon)$ such that

$\|\
(q-\Delta_{f}(n_{f}+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N^{[L(q,\epsilon)]}_{Q_{i}({\bf
f})}(r).$

(b) If the image of $f$ is contained in a subvariety $V$ of
${\mathbb{P}}^{N}({\mathbb{C}})$, then $1\leq n_{f}\leq\dim V$ and by Remark
3.5 we have $\Delta_{f}\leq(\dim V-n_{f}+1)\Delta_{V},$ and hence

$\|\ (q-\Delta_{V}(\dim
V-n_{f}+1)(n_{f}+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N^{[L(q,\epsilon)]}_{Q_{i}({\bf
f})}(r).$

Since $(\dim V-n_{f}+1)(n_{f}+1)\leq\left(\dfrac{\dim V}{2}+1\right)^{2}$, we
get the second main theorem

$\biggl{\|}\ \left(q-\Delta_{V}\left(\dfrac{\dim
V}{2}+1\right)^{2}-\epsilon\right)T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N^{[L(q,\epsilon)]}_{Q_{i}({\bf
f})}(r).$

(c) With the help of Lemma 3.8, using the construction method for the new
filtralation of G. Dethloff and T. V. Tan [4] (see also [20, Remark 3.1]) and
repeating the routine way in the proof of second main theorem, we may get the
following theorem without the condition on the algebraic degree of $f$.

######  Theorem 4.14.

Let $f$ be a nonconstant meromorphic map of $\mathbf{C}^{m}$ into
${\mathbb{P}}^{N}(\mathbf{C})$ with a reduced representation ${\bf
f}=(f_{0},\ldots,f_{N})$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be a family of slowly
(with respect to $f$) moving hypersurfaces in ${\mathbb{P}}^{N}({\mathbb{C}})$
with $\deg Q_{i}=d_{i}\ (1\leq i\leq q).$ Let $n_{f}$ be the algebraic
dimension of $f$ over $\mathcal{K}_{\\{Q_{i}\\}_{i=1}^{q}}$ and let
$\Delta_{f}$ be the distributive constant of $\\{Q_{i}\\}_{i=1}^{q}$ with
respect to $f$. Then for any $\epsilon>0$, we have

$\|\
(q-\Delta_{f}(n_{f}+1)-\epsilon)T_{f}(r)\leq\sum_{i=1}^{q}\frac{1}{d}N_{Q_{i}({\bf
f})}(r).$

However, by this method, the truncation level $[L(q,\epsilon)]$ can not be
given.

Disclosure statement: The author states that there is no conflict of interest.

Acknowledgments. The author would like to thank the referees for their helpful
comments and suggestions on the first version of this paper. This research is
funded by Vietnam National Foundation for Science and Technology Development
(NAFOSTED) under the grant number 101.02-2021.12.

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