Difference analogue of second main theorems for meromorphic mapping into algebraic variety

TL;DR

This paper develops difference analogues of second main theorems for meromorphic mappings into algebraic varieties, extending classical value distribution theory to difference settings and applications to uniqueness and degeneracy of holomorphic curves.

Contribution

It introduces difference analogues of second main theorems for meromorphic mappings into algebraic varieties, including results on degeneracy, Picard-type theorems, and uniqueness sharing hypersurfaces.

Findings

Established difference second main theorems for meromorphic mappings.

Proved algebraic degeneracy results for holomorphic curves intersecting hypersurfaces.

Derived a difference analogue of Picard's theorem and uniqueness theorems.

Abstract

In this paper, we prove some difference analogue of second main theorems of meromorphic mapping from Cm into an algebraic variety V intersecting a finite set of fixed hypersurfaces in subgeneral position. As an application, we prove a result on algebraically degenerate of holomorphic curves intersecting hypersurfaces and difference analogue of Picard's theorem on holomorphic curves. Furthermore, we obtain a second main theorem of meromorphic mappings intersecting hypersurfaces in N-subgeneral position for Veronese embedding in Pn(C) and a uniqueness theorem sharing hypersurfaces.

Equations

$${\nu }^M(z)=\min \{\nu (z),M\},{\nu }_{\le k}^M(z)=\{\begin{array}{ll}& 0\text{if}\nu (z)>k\\ & {\nu }^M(z)\text{if}\nu (z)\le k\end{array}$$

$${\nu }_{\ge k}^M(z)=\{\begin{array}{ll}& 0\text{if}\nu (z)<k\\ & {\nu }^M(z)\text{if}\nu (z)\ge k\end{array},$$

$$n(t)=\{\begin{array}{ll}& {\int }_{\mathrm{supp}\nu \cap B(t)}\nu v\text{if}m\ge 2\\ & {\sum }_{|z|\le t}\nu (z)\text{if}m=1\end{array}.$$

$$N(r,\nu )=\underset{1}{\overset{r}{\int }}\frac{n(t)}{t^{2m-1}}dt$$

$$N_h(r)=N(r,{\nu }_h),N_h^M(r)=N(r,{\nu }_h^M),$$

$$N_{h,\le k}^M(r)=N(r,{\nu }_{h,\le k}^M),N_{h,\ge k}^M(r)=N(r,{\nu }_{h,\ge k}^M).$$

$$T_f(r)=\underset{S(r)}{\int }\log ||\tilde{f}(z)||\sigma (z)-\underset{S(1)}{\int }\log ||\tilde{f}(z)||\sigma (z),$$

$$\varsigma (f)=\underset{r\to \infty }{\mathrm{limsup}}\frac{\log \log T_f(r)}{\log r},\sigma (f)=\underset{r\to \infty }{\mathrm{limsup}}\frac{\log T_f(r)}{\log r}.$$

$$S(r,f)=o(T_f(r))$$

$$L(x_0,\dots ,x_n)=L_{\mathbf{a}}(x_0,\dots ,x_n)=\sum _{j=0}^n{a}_j{x}_j$$

$$H=H_{\mathbf{a}}=\{x=(x_0:\cdots :x_n)\in {\mathbb{P}}^n(\mathbb{C})|L_{\mathbf{a}}(x)=L_{\mathbf{a}}(x_0,\dots ,x_n)=0\}.$$

$$m_f(r,H)=\underset{S(r)}{\int }\log \frac{||\tilde{f}(z)||\cdot ||L_{\mathbf{a}}||}{|L_{\mathbf{a}}(\tilde{f}(z))|}\sigma (z)-\underset{S(1)}{\int }\log \frac{||\tilde{f}(z)||\cdot ||L_{\mathbf{a}}||}{|L_{\mathbf{a}}(\tilde{f}(z))|}\sigma (z)$$

$$D=\{x=(x_0:\cdots :x_n)\in {\mathbb{P}}^n(\mathbb{C})|Q(x)=Q(x_0,\dots ,x_n)=0\}$$

$$m_f(r,D)=\underset{S(r)}{\int }\log \frac{||\tilde{f}(z)|{|}^d||Q||}{|Q(\tilde{f}(z))|}\sigma (z)-\underset{S(1)}{\int }\log \frac{||\tilde{f}(z)|{|}^d||Q||}{|Q(\tilde{f}(z))|}\sigma (z)$$

$$N_f(r,D)=N(r,{\nu }_{Q(\tilde{f})}),N_f^M(r,D)=N(r,{\nu }_{Q(\tilde{f})}^M),$$

$$N_{f,\le k}^M(r,D)=N(r,{\nu }_{Q(\tilde{f}),\le k}^M),N_{f,\ge k}^M(r,D)=N(r,{\nu }_{Q(\tilde{f}),\ge k}^M)$$

$${\nu }_{h-a}={\nu }_{h-a}^{[M,c]}+{\tilde{\nu }}_{h-a}^{[M,c]}$$

$$\nu_{h-a}^{[M,c]}(z)=\left\{\begin{array}[]{ll}\nu_{h-a}(z),&\hbox{ if $\nu_{h_{jc}-a}(z)\geq\nu_{h-a}(z)$ for $j=1,...,M$;}\\ 0,&\hbox{otherwise,}\end{array}\right.$$

$${\nu }_{Q(\tilde{f})}={\nu }_{Q(\tilde{f})}^{[M,c]}+{\tilde{\nu }}_{Q(\tilde{f})}^{[M,c]}$$

$${\stackrel{\sim }{N}}_f^{[M,c]}(r,D)={\stackrel{\sim }{N}}_{Q(\tilde{f})}^{[M,c]}(r).$$

$${\nu }_{Q(\tilde{f})}={\dot{\nu }}_{Q(\tilde{f})}^{[M,p]}+{\hat{\nu }}_{Q(\tilde{f})}^{[M,p]}$$

$$\dot{\nu}_{Q(\tilde{f})}^{[M,p]}(z)=\left\{\begin{array}[]{ll}\nu_{Q(\tilde{f})}(z),&\hbox{ if $\nu_{Q(\tilde{f})_{p^{j}}}(z)\geq\nu_{Q(\tilde{f})}(z)$ for $j=1,\dots,M$;}\\ 0,&\hbox{otherwise,}\end{array}\right.$$

$$(q-N_k-\epsilon )T_f(r)$$

$$V\cap \text{Supp}{D}_{i_1}\cap \cdots \cap \text{Supp}{D}_{i_{N+1}}=\varnothing ,$$

$$\sum _{j=1}^q{\stackrel{\sim }{\delta }}_f^{[M,c]}(r,D_j)\le (N-k+1)(k+1)+\epsilon ,$$

$$Q(\mathbf{x})=\sum _{j=0}^{n_d}{a}_j{\mathbf{x}}^{I_j},$$

$${\varrho }_d(x)=(w_0(x):\cdots :w_{n_{\mathcal{D}}}(x))=({\mathbf{x}}^{I_0}:\cdots :{\mathbf{x}}^{I_{n_d}}),$$

$$L(w)=a_0{w}_0+\cdots +a_{n_d}{w}_{n_d}$$

$$C_{\varrho_{d}}(f(z))=\left|\begin{array}[]{cccc}{\tilde{f}}^{I_{0}}(z)&{\tilde{f}}^{I_{1}}(z)&\cdots&{\tilde{f}}^{I_{n_{d}}}(z)\\ {\tilde{f}}^{I_{0}}(z+{c})&{\tilde{f}}^{I_{1}}(z+{c})&\cdots&{\tilde{f}}^{I_{n_{d}}}(z+{c})\\ \vdots&\vdots&\ddots&\vdots\\ {\tilde{f}}^{I_{0}}(z+n_{d}{c})&{\tilde{f}}^{I_{1}}(z+n_{d}{c})&\cdots&{\tilde{f}}^{I_{n_{d}}}(z+n_{d}{c})\\ \end{array}\right|.$$

$$C(f(z)):=C_{\varrho_{1}}(f(z))=\left|\begin{array}[]{cccc}f_{0}(z)&f_{1}(z)&\cdots&f_{n}(z)\\ f_{0}(z+{c})&f_{1}(z+{c})&\cdots&f_{n}(z+{c})\\ \vdots&\vdots&\ddots&\vdots\\ f_{0}(z+n{c})&f_{1}(z+n{c})&\cdots&f_{n}(z+n{c})\\ \end{array}\right|$$

Full text

2010 Mathematics Subject Classification. Primary 32H30.

Key words: Algebraic variety, Meromorphic mapping, Nevanlinna theory.

††The research results are sponsored by China/Shandong University International Postdoctoral Exchange Program, NSFC of Shandong (No. ZR2018MA014), PCSIRT (No. IRT1264) and The Fundamental Research Funds of Shandong University(No. 2017JC019).

1. Introduction and main results

Recently, the second main theorem of Nevanlinna have been studied actively for difference operators. For example, R. Halburd and R. Korhonen [6, 7] in 2006 built the second main theorem for difference operators of meromorphic functions on $\mathbb{C}$. In 2014, R. Halburd, R. Korhonen and K. Tohge [8] proved the difference analogue of second main theorem of holomorphic curves from $\mathbb{C}$ into $\mathbb{P}^{n}(\mathbb{C})$ intersecting a finite set of fixed hyperplanes in general position.

In 2009, M. Ru [16] proved the second main theorem of holomorphic curves into an algebraic variety. In 2017, S. D. Quang [14] extended the result of M. Ru [16] to hypersurfaces in subgeneral position. Our goal is to establish the difference analogue of second main theorem of meromorphic mappings from $\mathbb{C}^{m}$ into an algebraic variety $V$ intersecting a finite set of fixed hypersurfaces in subgeneral position depending on a step number of difference. In particular, for the Veronese embedding in $\mathbb{P}^{n}(\mathbb{C}),$ our second main theorem and difference analogue of Picard’s theorem recover the results of Cao-Korhonen [2] and Halburd-Korhonen-Tohge [8], respectively. By a way, we also obtain uniqueness theorems of meromorphic mappings which improve the result of Dulock-Ru [4].

To introduce our results clearly, it is necessary to introduce some notations. Take $z=(z_{1},\dots,z_{m})\in\mathbb{C}^{m}$ and write the standard norm $||z||=(\sum_{j=1}^{m}|z_{j}|^{2})^{1/2}$. For $r>0,$ define the ball $B(r)=\{z\in\mathbb{C}^{m}:||z||<r\}$ and the sphere $S(r)=\{z\in\mathbb{C}^{m}:||z||=r\}$. As usual, define the differential operator $d^{c}=\dfrac{1}{4\pi i}(\partial-\overline{\partial})$ and two differential forms $v(z)=(dd^{c}||z||^{2})^{m-1}$, $\sigma(z)=d^{c}\log||z||^{2}\wedge(dd^{c}||z||^{2})^{m-1}.$

Take $z_{0}\in\mathbb{C}^{m}$ and $a\in\mathbb{P}^{1}(\mathbb{C})$. We expand a nonzero entire function $h$ on $\mathbb{C}^{m}$ into a Taylor series $h(z)=\sum_{n=0}^{\infty}P_{n}(z-z_{0})$ at $z_{0}$, where $P_{n}(z)$ is either identically zero or a homogeneous polynomial of $z$ with degree $n.$ The number $\nu_{h}(z_{0})=\mathop{\mathrm{min}}\{n:P_{n}\neq 0\}$ is said to be the zero multiplicity of $h$ at $z_{0}.$ Set ${\rm supp}\nu_{h}:=\overline{\{z\in\mathbb{C}^{m}:\nu_{h}(z)\neq 0\}},$ which is a purely $(m-1)$-dimensional analytic subset or empty set. Further, if $h$ is a nonzero meromorphic function on $\mathbb{C}^{m}$, we can choose two holomorphic functions $h_{0}$ and $h_{1}$ on a neighborhood $U$ of $z_{0}$ such that $h=\dfrac{h_{0}}{h_{1}}$ on $U$ and $\dim h_{0}^{-1}(0)\cap h_{1}^{-1}(0)\leq m-2$. Define the $a$-valued multiplicity of $h$ at $z_{0}$ by $\nu_{h-a}(z_{0})=\nu_{h=a}(z_{0}):=\nu_{h_{0}-ah_{1}}(z_{0})$, where particularly $\nu_{h}=\nu_{h=0}=\nu_{h_{0}},\nu_{h=\infty}=\nu_{h_{1}},$ which are independent of the choices of $h_{0}$ and $h_{1}.$

In Nevanlinna’s theory, a multiplicity function (or divisor) $\nu$ on $\mathbb{C}^{m}$ usually be associated with the following truncation functions

[TABLE]

and

[TABLE]

where $k,M$ are positive integers or $+\infty.$ Further, the function $\nu$ defines a counting function as follows:

[TABLE]

Similarly, the truncation functions $\nu^{M}$, $\nu_{\leq k}^{M}$ and $\nu_{\geq k}^{M}$ can define counting functions $n^{M}(t),n_{\leq k}^{M}(t)$ and $n_{\geq k}^{M}(t)$, respectively. Moreover, the function $\nu$ also defines a valence function

[TABLE]

for $r>1$, so that the valence functions $N(r,\nu^{M}),N(r,\nu_{\leq k}^{M})$, $N(r,\nu_{\geq k}^{M})$ are associated with the truncation functions $\nu^{M}$, $\nu_{\leq k}^{M}$ and $\nu_{\geq k}^{M}$, respectively, which also be denoted by $N^{M}(r,\nu),$ $N_{\leq k}^{M}(r,\nu)$ and $N_{\geq k}^{M}(r,\nu),$ respectively. In particular, for a meromorphic function $h$ on $\mathbb{C}^{m},$ we write

[TABLE]

[TABLE]

Let $f:\mathbb{C}^{m}\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a nonconstant meromorphic mapping. We can choose holomorphic functions $f_{0},f_{1},\dots,f_{n}$ on $\mathbb{C}^{m}$ such that $I_{f}:=\{z\in\mathbb{C}^{m}:f_{0}(z)=\dots=f_{n}(z)=0\}$ is of dimension at most $m-2$ and $f=(f_{0}:\dots:f_{n}).$ Usually, $\tilde{f}=(f_{0},\dots,f_{n}):\mathbb{C}^{m}\longrightarrow\mathbb{C}^{n+1}$ is called a reduced representation of $f.$ For $r>1$, the characteristic function of $f$ can be given by

[TABLE]

where $||\tilde{f}(z)||=(\sum_{j=0}^{n}|f_{j}(z)|^{2})^{1/2}.$ Further, we denote the hyper-order and order of $f$ respectively by

[TABLE]

Moreover, we use $S(r,f)$ (resp., $S_{1}(r,f)$) to denote a quantity satisfying the following property

[TABLE]

outside a set of finite logarithmic (or Lebesgue) measure (resp., $S_{1}(r,f)=o(T_{f}(r))$ on a set of logarithmic density $1$).

Take ${\bf a}=(a_{0},\dots,a_{n})\in\mathbb{C}^{n+1}$ and define a linear form

[TABLE]

associated with a hyperplane

[TABLE]

If $f(\mathbb{C}^{m})\not\subset H,$ the proximity function of $f$ with respective to $H$ can be obtained by

[TABLE]

for $r>1$, where $||L_{\bf a}||=(\sum_{j=0}^{n}|a_{j}|^{2})^{1/2}$, in which, up to an additive constant, we may choose $||\tilde{f}(z)||=\max\{|f_{0}(z)|,\dots,|f_{n}(z)|\}$ and $||L_{\bf a}||=\sum_{j=0}^{n}|a_{j}|$ since norms on $\mathbb{C}^{m}$ are equivalent.

Generally, if $Q(x_{0},\dots,x_{n})\in\mathbb{C}[x_{0},...,x_{n}]$ is a homogeneous polynomial of degree $d$, then a hypersurface

[TABLE]

of degree $d$ in $\mathbb{P}^{n}(\mathbb{C})$ is associated with $Q$. If $f(\mathbb{C}^{m})\not\subset D,$ i.e., $Q(\tilde{f})\not\equiv 0,$ the proximity function $m_{f}(r,D)$ of $f$ can be given by

[TABLE]

for $r>1$, where $||Q||$ is the total of the absolute values of the coefficients of $Q.$ Further, let $\nu_{Q(\tilde{f})}$ be the zero multiplicity function of $Q(\tilde{f})$. Then the following valence functions

[TABLE]

[TABLE]

are defined well. We also use the symbol $\overline{E}_{M)}(D,f)$ to denote ${\rm supp}\nu_{Q(\tilde{f}),\leq M}^{M}$, that is, $\overline{E}_{M)}(D,f)$ is the set of zeros of $Q(\tilde{f})$ with multiplicity at most $M,$ in which each zero is counted only one time.

Our work is based on a decomposition of the $a$-valued multiplicity $\nu_{h-a}$ of a meromorphic function $h$ on $\mathbb{C}^{m}$

[TABLE]

defined by R. Korhonen, N. Li and K. Tohge in [10], in which $M$ is a positive integer, $c\in\mathbb{C}^{m}\setminus\{0\}$, and

[TABLE]

where $h_{jc}(z)=h(z+jc)$. The points on ${\rm supp}\nu_{h-a}^{[M,c]}$ (resp., ${\rm supp}\widetilde{\nu}_{h-a}^{[M,c]}$) are called to be $M-$ successive and $c-$separated (resp., $M$-aperiodic of pace $c$), which defines the valence function $N_{h-a}^{[M,c]}(r)$ (resp., $\overset{\sim}{N}_{h-a}^{[M,c]}(r)$). It is obvious that the number $M$ serves as the step of difference.

Thus the zero multiplicity function $\nu_{Q(\tilde{f})}$ of $Q(\tilde{f})$ defined by the hypersurface $D$ in $\mathbb{P}^{n}(\mathbb{C})$ of degree $d$ corresponds Korhonen-Li-Tohge’s decomposition

[TABLE]

associated with valence functions $N_{f}^{[M,c]}(r,D)$ and $\overset{\sim}{N}_{f}^{[M,c]}(r,D)$ of $\nu_{Q(\tilde{f})}^{[M,c]}$ and $\widetilde{\nu}_{Q(\tilde{f})}^{[M,c]}$, respectively, and hence

[TABLE]

Similarly, we also consider the following decomposition

[TABLE]

in which $p=(p_{1},\dots,p_{m})\in\mathbb{C}^{m}\setminus\{0,1\}$, and

[TABLE]

where $Q(\tilde{f})_{p^{j}}(z)=Q(\tilde{f})(p^{j}\cdot z)=Q(\tilde{f})(p_{1}^{j}z_{1},\dots,p_{m}^{j}z_{m})$, and denote valence functions $\dot{N}_{f}^{[M,p]}(r,D)$ and $\hat{N}_{f}^{[M,p]}(r,D)$ of $\dot{\nu}_{Q(\tilde{f})}^{[M,p]}$ and $\hat{\nu}_{Q(\tilde{f})}^{[M,p]}$, respectively. Now we can state first result as follows:

Theorem 1**.**

Fix $c\in\mathbb{C}^{m}\setminus\{0\}$ and take positive integers $q,N,k$ with $q>(N-k+1)(k+1),N\geq k$. Let $V$ be a complex projective variety of dimension $k$ embedding into $\mathbb{P}^{n}(\mathbb{C})$. Let $D_{j}\ (1\leq j\leq q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position on $V.$ Let $d$ be the least common multiple of $d_{1},\dots,d_{q}.$ Let $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\to V$ be a algebraically non-degenerate meromorphic mapping on $\mathcal{P}^{1}_{c}$ with $\varsigma(f)=\varsigma<1.$ Then for any $\varepsilon>0$, we have

[TABLE]

where $N_{k}=(N-k+1)(k+1),$ M=\dfrac{(dk)^{k}\deg V}{k!}\Big{(}1+2ld^{k}\deg V(N-k+1)(2k+1)I(\varepsilon^{-1})\Big{)}^{k}, in which $I(x):=\mathop{\mathrm{min}}\{k\in\mathbb{N}:k>x\}$ for $x>0.$

Recall that the hypersurfaces $D_{1},\dots,D_{q}$ are said to be in $N$-subgeneral position on V if for every subset $\{i_{1},\dots,i_{N+1}\}\subset\{1,\dots,q\},$ we have

[TABLE]

where $\text{Supp}(D)$ means the support of the divisor $D.$ If $N=k,$ the hypersurfaces $D_{1},\dots,D_{q}$ are said to be in general position on V. In Theorem 1, $\mathcal{P}^{1}_{c}$ is the field of meromorphic functions $h$ of period $c$ in $\mathbb{C}^{m}$ with the hyper-order $\varsigma(h)<1$. Suppose that $V$ is generated by a homogeneous ideal $\mathcal{I}(V)$ in $\mathbb{C}[x_{0},\dots,x_{n}]$ and let $\mathcal{I}_{\mathcal{P}^{1}_{c}}(V)$ be the ideal in $\mathcal{P}^{1}_{c}[x_{0},\dots,x_{n}]$ generated by $\mathcal{I}(V),$ so that $P(f_{0},\dots,f_{n})=0$ for all $P\in\mathcal{I}_{\mathcal{P}^{1}_{c}}(V).$ The mapping $f$ is said to be algebraically nondegenerate over $\mathcal{P}^{1}_{c}$ if there is not $Q\in\mathcal{P}^{1}_{c}[x_{0},\dots,x_{n}]\setminus\mathcal{I}_{\mathcal{P}^{1}_{c}}(V)$ such that $Q(f_{0},\dots,f_{n})=0.$ When $V=\mathbb{P}^{n}(\mathbb{C}),$ $f$ is algebraically nondegenerate over $\mathcal{P}^{1}_{c}$ if there is not a nonzero homogeneous polynomial $Q\in\mathcal{P}^{1}_{c}[x_{0},\dots,x_{n}]$ such that $Q(f_{0},\dots,f_{n})=0.$

Theorem 1 is a difference analogue of second main theorem due to S. D. Quang [14], which implies the following defect relation immediately:

[TABLE]

where $\overset{\sim}{\delta}_{f}^{[M,c]}(r,D_{j})=1-\limsup_{r\to\infty}\dfrac{\overset{\sim}{N}_{f}^{[M,c]}(r,D_{j})}{d_{j}T_{f}(r)}.$ When $N=k,$ it is a difference analogue of the defect relation due to M. Ru [16].

Next we show that by using Veronese imbedding, the second main theorem can be modified such that the step number $M$ of difference does not depend on $\varepsilon.$ Let $D$ be a hypersurface of degree $d$ in $\mathbb{P}^{n}(\mathbb{C})$, which is defined by a homogeneous polynomial $Q$ of degree $d$

[TABLE]

where ${\bf x}=(x_{0},\dots,x_{n})\in\mathbb{C}^{n+1}$, $I_{j}=(i_{j0},\dots,i_{jn})\in\mathbb{Z}_{+}^{n+1}$ is a $(n+1)$-fold index over the set $\mathbb{Z}_{+}$ of nonnegative integers with $|I_{j}|=i_{j0}+\dots+i_{jn}=d$ for $j=0,\dots,n_{d}$, $n_{d}=\binom{n+d}{n}-1$, ${\bf a}=(a_{0},\dots,a_{n_{d}})\in\mathbb{C}^{n_{d}+1}$, ${\bf x}^{I_{j}}=x_{0}^{i_{j0}}\dots x_{n}^{i_{jn}}$, and where $I_{0},\dots,I_{n_{d}}$ are lexicographic ordering. Let $w=(w_{0}:\dots:w_{n_{d}})$ be a homogeneous coordinate in $\mathbb{P}^{n_{d}}(\mathbb{C})$ and let $\varrho_{d}:\mathbb{P}^{n}(\mathbb{C})\to\mathbb{P}^{n_{d}}(\mathbb{C})$ be the Veronese embedding of degree $d$ defined by

[TABLE]

where $x=(x_{0}:\dots:x_{n})$ is a homogeneous coordinate in $\mathbb{P}^{n}(\mathbb{C})$. Then a linear form

[TABLE]

and a hyperplane $H=\{w:L(w)=0\}$ in $\mathbb{P}^{n_{d}}(\mathbb{C})$ are associated with $D$ (or $Q$). By a way, Casorati determinant of the mapping $f:\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ respect to Veronese embedding $\varrho_{d}$ and ${c}\in\mathbb{C}^{m}\setminus\{0\}$ is defined by

[TABLE]

When $d=1,$ we call

[TABLE]

by the Casorati determinant of $f.$

Let $\mathcal{D}=\{D_{1},\dots,D_{q}\}$ be a collection of arbitrary hypersurfaces and let $Q_{j}$ be the homogeneous polynomial of degree $d_{j}$ in $\mathbb{C}[x_{0},\dots,x_{n}]$ defining $D_{j}$ for $j=1,\dots,q$. Let $d=d_{\mathcal{D}}$ be the least common multiple of $d_{1},...,d_{q}$ and set

[TABLE]

Set $Q^{*}_{j}=Q_{j}^{d/d_{j}}$, so that a vector ${\bf a}^{*}_{j}\in\mathbb{C}^{n_{\mathcal{D}}+1}$ and a hyperplane $H_{j}$ in $\mathbb{P}^{n_{\mathcal{D}}}(\mathbb{C})$ are associated with $Q^{*}_{j}$ for $j=1,\dots,q.$ If $q>n_{\mathcal{D}}$, $N\geq n_{\mathcal{D}}$, the collection $\mathcal{D}$ is said to be in $N$-subgeneral position for Veronese embedding in $\mathbb{P}^{n}(\mathbb{C})$ if $\{H_{1},\dots,H_{q}\}$ is in $N$-subgeneral position in $\mathbb{P}^{n_{\mathcal{D}}}(\mathbb{C})$, that is, for any distinct indices $i_{0},\dots,i_{N}$ lied in $\{1,\dots,q\}$, the vectors ${\bf a}^{*}_{i_{0}},\dots,{\bf a}^{*}_{i_{N}}$ have rank $n_{\mathcal{D}}.$

Theorem 2**.**

Let $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be algebraically non-degenerate meromorphic mapping. Take an integer $N$ with $N\geq n_{\mathcal{D}}$ and let $D_{j}$ $(1\leq j\leq q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position for Veronese embedding. Let $d$ be the least common multiple of $d_{1},\dots,d_{q}$. If $q>2N-n_{\mathcal{D}}+1$, then for any $1<r<+\infty,$

[TABLE]

Theorem 3**.**

Take an integer $N$ with $N\geq n_{\mathcal{D}}$ and let $D_{j}\ (1\leq j\leq q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position for Veronese embedding. Let $d$ be the least common multiple of $d_{1},\dots,d_{q}$ and set $d^{\prime}=\mathop{\mathrm{min}}_{j=1}^{q}\{d_{j}\}.$ Let $f,g:\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be algebraically non-degenerate meromorphic mappings satisfying the following conditions:

**(a): **

$\dim f^{-1}(D_{i})\cap f^{-1}(D_{j})\leq m-2$* and $\dim g^{-1}(D_{i})\cap g^{-1}(D_{j})\leq m-2$ for all $1\leq i<j\leq q;$*

**(b): **

there exist positive integers $m_{j}$ with $m_{1}\geq\dots\geq m_{q}$ such that

[TABLE]

and $f(z)=g(z)$ for $z\in\cup_{j=1}^{q}\overline{E}_{m_{j})}(D_{j},f)$.

Then $f\equiv g$ if

[TABLE]

If we take $N=n_{\mathcal{D}}$ and let $m_{j}\to\infty$ for each $j$ in Theorem 3, we get a uniqueness theorem under $q\geq 3n_{\mathcal{D}}+2.$ The number of hypersurfaces in Theorem 2 is smaller than that appeared in the result of Dulock-Ru [4].

Theorem 4**.**

Let $D_{j}\ (1\leq j\leq q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position for Veronese embedding. Let $d$ be the least common multiple of $d_{1},\dots,d_{q}$. Let $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be algebraically non-degenerate meromorphic mapping on $\mathcal{P}^{1}_{c}$ with $\varsigma(f)<1.$ If $q>2N-n_{\mathcal{D}}+1,$ $N\geq n_{\mathcal{D}}$, we have

[TABLE]

When $d=1,$ then $n_{D}=n,$ Theorem 4 is just the difference analogue of the second main theorem of meromorphic mappings due to Cao-Korhonen [2].

Corollary 1**.**

Take $q=n_{\mathcal{D}}+2$ and let $D_{j}\ (1\leq j\leq q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position for Veronese embedding. Let $d$ be the least common multiple of $d_{1},\dots,d_{q}.$ Then a nonconstant meromorphic mapping $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ is algebraically degenerate on $\mathcal{P}^{1}_{c}$ if $\varsigma(f)<1$ and if

[TABLE]

If the preimages of hypersurfaces $D_{j}\ (j=1,\dots,n_{\mathcal{D}}+2)$ under $f$ are forward invariant with respect to the translation $\tau(z)=z+c,$ that is, $\tau(f^{-1}(D))\subset f^{-1}(D)$ counting multiplicity, then Corollary 1 means that $f$ is algebraically degenerate on $\mathcal{P}^{1}_{c}.$

Theorem 5**.**

Take $c\in\mathbb{C}^{m}\setminus\{0\}$ and let $f:\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be a meromorphic mapping with $\varsigma(f)<1.$ Let $D_{j}\ (j=1,\dots,q)$ be a hypersurface of degree $d_{j}$ in $P^{n}(\mathbb{C})$ such that $D_{1},...,D_{q}$ are in $N$-subgeneral position for Veronese embedding. Let $d$ be the least common multiple of $d_{1},\dots,d_{q}.$ If the preimages of hypersurfaces $D_{j}$ under $f$ are forward invariant with respect to the translation $\tau(z)=z+c,$ then the image of $f$ is contained in one of hypersurfaces $D_{j}$ or the image of $\varrho_{d}(f)$ is contained in a projective linear subspace of dimension $\leq\left[\dfrac{N}{q-N}\right]$ over $\mathcal{P}_{c}^{1}$. In particular, we have $f=f_{c}:=f\circ\tau$ if $q\geq 2N+1$.

When $d=1,$ then $n_{D}=n.$ If we choose $N=n,q=n+p$ in Theorem 5, it is a difference analogue of Picard’s theorem for holomorphic curves due to Halburd-Korhonen-Tohge [8].

Take $p=(p_{1},\dots,p_{m})\in\mathbb{C}^{m}\setminus\{0,1\}$ and write $p\cdot z=(p_{1}z_{1},\dots,p_{m}z_{m})$ for $z=(z_{1},\dots,z_{m})$. Let $\mathcal{P}^{0}_{p}$ be the field of meromorphic functions $h$ of zero order in $\mathbb{C}^{m}$ such that $h(p\cdot z)=h(z).$ Similarly, the meromorphic mapping $f:\mathbb{C}^{m}\to V\subset\mathbb{P}^{n}(\mathbb{C})$ is said to be algebraically nondegenerate over $\mathcal{P}^{0}_{p}$ if there is not $Q\in\mathcal{P}_{p}^{0}[x_{0},\dots,x_{n}]\setminus\mathcal{I}_{\mathcal{P}^{0}_{p}}(V)$ such that $Q(f_{0},\dots,f_{n})=0$, where $\mathcal{I}_{\mathcal{P}^{0}_{p}}(V)$ is the ideal in $\mathcal{P}^{0}_{p}[x_{0},\dots,x_{n}]$ generated by $\mathcal{I}(V).$ When $V=\mathbb{P}^{n}(\mathbb{C}),$ $f$ is algebraically nondegenerate over $\mathcal{P}^{0}_{p}$ if there is not a polynomial homogeneous $Q\in\mathcal{P}^{0}_{p}[x_{0},\dots,x_{n}]\setminus\{0\}$ such that $Q(f_{0},\dots,f_{n})=0.$ By a way, the $p$-Casorati determinant of a meromorphic mapping $f:\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ is defined by

[TABLE]

in which $p^{j}=(p_{1}^{j},\dots,p_{m}^{j})$. By a result in [3], we know that $C_{p}(f)\equiv 0$ if and only if $f_{0},\dots,f_{n}$ are linear dependent over the filed $\mathcal{P}^{0}_{p}.$ By using the $p$-difference analogue of logarithmic derivative lemma (cf.[13] and [3]), we can obtain the following results for $p$-difference operator without proofs.

Theorem 6**.**

Take positive integers $N,k$ with $N\geq k$ and let $V\subset\mathbb{P}^{n}(\mathbb{C})$ be a complex projective variety of dimension $k\geq 1.$ Let $D_{j}\ (1\leq j\leq q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position on $V.$ Let $d$ be the least common multiple of $d_{1},\dots,d_{q}.$ Let $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\to V$ be algebraically non-degenerate meromorphic mapping of zero order on $\mathcal{P}^{0}_{p}$. Assume that $q>N_{k}.$ Then for any $\varepsilon>0,$ we have

[TABLE]

where M=\dfrac{(dk)^{k}\deg V}{k!}\Big{(}1+2ld^{k}\deg V(N-k+1)(2k+1)I(\varepsilon^{-1})\Big{)}^{k}.

Theorem 7**.**

Let $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be algebraically non-degenerate meromorphic mapping of zero order on $\mathcal{P}^{0}_{p}$. Let $D_{j}\ (1\leq j\leq q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position for Veronese embedding. Let $d$ be the least common multiple of $d_{1},\dots,d_{q}$. If $q>2N-n_{\mathcal{D}}+1,$ $N\geq n_{\mathcal{D}}$, then we have

[TABLE]

Theorem 8**.**

Let $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be a meromorphic mapping of zero order. Let $D_{j}\ (j=1,\dots,q)$ be a hypersurface of degree $d_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ such that $D_{1},\dots,D_{q}$ are in $N$-subgeneral position for Veronese embedding. Let $d$ be the least common multiple of $d_{1},\dots,d_{q}.$ If the preimages of hypersurfaces $D_{j}$ under $f$ are forward invariant with respect to the rescaling $\tau(z)=p\cdot z,$ then the image of $f$ is contained in one of hypersurfaces $D_{j}$ or the image of $\varrho_{d}(f)$ is contained in a projective linear subspace of dimension $\leq\left[\dfrac{N}{q-N}\right]$ over $\mathcal{P}^{0}_{p}$. In particular, we have $f=f\circ\tau$ if $q\geq 2N+1$.

2. Some Lemmas

In order to prove theorems above, we need the following lemmas.

Lemma 1**.**

[8, 2]** Let $f$ be a non-constant meromorphic function in $\mathbb{C}^{m},$ and let ${c}\in\mathbb{C}^{m}\setminus\{0\}.$ If $\varsigma(f)<1,$ then the function $f_{c}(z)=f(z+c)$ satisfies

[TABLE]

From Lemma 1, if we repalce $f$ by $\dfrac{1}{f},$ and using First Main Theorem, we have $\varsigma\left(\dfrac{1}{f}\right)=\varsigma(f)<1,$ then

[TABLE]

Lemma 2**.**

[8, 2]** Take $c\in\mathbb{C}^{m}\setminus\{0\}$. If a meromorphic mapping $g=[g_{0}:\dots:g_{n}]:\mathbb{C}^{m}\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ satisfies $\varsigma(g)<1$, then the Casorati determinant of $g$ satisfies $C(g)\equiv 0$ if and only if the entire functions $g_{0},\dots,g_{n}$ are linearly dependent over the field $\mathcal{P}^{1}_{c}.$

Lemma 3**.**

[8, 2]** Take $c\in\mathbb{C}^{m}\setminus\{0\}$ and let $g=(g_{0}:\dots:g_{n}):\mathbb{C}^{m}\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a meromorphic mapping with $\varsigma(g)<1$ such that all zeros of $g_{0},\dots,g_{n}$ are forward invariant with respect to the translation $\tau(z)=z+c.$ Let $S_{1}\cup\dots\cup S_{l}$ be a partition of $\{0,\dots,n\}$ formed in such a way that $i$ and $j$ are the same class $S_{k}$ if and only if $g_{i}/g_{j}\in\mathcal{P}_{c}^{1}.$ If $g_{0}+\dots+g_{n}=0,$ then for each $k\in\{1,\dots,l\}$, we have

[TABLE]

Lemma 4**.**

[8]** Let $T:[0,+\infty)\to[0,+\infty)$ be a non-decreasing continuous function such that the hyper-order of $T$ is strictly less than one, that is,

[TABLE]

If $\delta\in(0,1-\varsigma),$ $s\in(0,\infty)$, then

[TABLE]

where $r$ runs to infinity outside of a set of finite logarithmic measure.

Take ${\bf c}=(c_{0},\dots,c_{n})\in\mathbb{R}^{n+1}$. Let $V\subset\mathbb{P}^{n}(\mathbb{C})$ be a projective variety of dimension $k$ and let

[TABLE]

be the Chow form associated to $V$, that is, $F_{V}$ is irreducible in $\mathbb{C}[x_{00},\dots,x_{kn}],$ $F_{V}$ is a homogeneous polynomial of degree $\Delta=\deg V$ in each block ${\bf x}_{i}=(x_{i0},\dots,x_{in})$ $(0\leq i\leq k)$ and $F_{V}({\bf x}_{0},\dots,{\bf x}_{k})=0$ if and only if $X\cap H_{{\bf x}_{0}}\cap\dots\cap H_{{\bf x}_{k}}\neq\emptyset,$ where $H_{{\bf x}_{i}}$ is a hyperplane given by

[TABLE]

For an auxiliary variable $t$, we obtain a decomposition

[TABLE]

where $G_{0},\dots,G_{r}\in\mathbb{C}[x_{00},\dots,x_{0n};\dots;x_{k0},\dots,x_{kn}]$ and $e_{0}>e_{1}>\dots>e_{r}.$ The Chow weight of $V$ with respect to ${\bf c}$ is defined by

[TABLE]

Suppose that $V$ is generated by a prime ideal $\mathcal{I}(V)$ in $\mathbb{C}[x_{0},\dots,x_{n}]$ and let $\mathbb{C}[x_{0},\dots,x_{n}]_{u}$ be the vector space of homogeneous polynomials in $\mathbb{C}[x_{0},\dots,x_{n}]$ of degree $u$ (including 0). Put $\mathcal{I}_{u}(V):=\mathbb{C}[x_{0},\dots,x_{n}]_{u}\cap\mathcal{I}(V)$ and denote the Hilbert function $H_{V}$ of $V$ by

[TABLE]

We define the $u$-th Hilbert weight $S_{V}(u,{\bf c})$ of $V$ with respect to ${\bf c}$ by

[TABLE]

where the maximum is taken over all sets of monomials ${\bf x}^{I_{1}},\dots,{\bf x}^{I_{H_{V}(u)}}$ whose residue classes module $\mathcal{I}(V)$ form a basis of $\mathbb{C}[x_{0},\dots,x_{n}]_{u}/\mathcal{I}_{u}(V)$, in which $I_{j}=(i_{j0},\dots,i_{jn})\in\mathbb{Z}_{+}^{n+1}$ is a $(n+1)$-fold index.

Lemma 5**.**

[16]** Take ${\bf c}=(c_{0},\dots,c_{n})\in\mathbb{R}^{n+1}$ with $c_{j}\geq 0$ for $j=0,...,n$. Let $V\subset\mathbb{P}^{n}(\mathbb{C})$ be an algebraic variety of dimension $k$. If $u>\Delta=\deg V$, then

[TABLE]

Lemma 6**.**

[5, 16]** Take ${\bf c}=(c_{0},\dots,c_{n})\in\mathbb{R}^{n+1}$ with $c_{j}\geq 0$ for $j=0,\dots,n$. Let $V$ be a subvariety of $\mathbb{P}^{n}(\mathbb{C})$ of dimension $k$ and let $\{i_{0},\dots,i_{k}\}$ be a subset of $\{0,\dots,n\}$ such that $\{x_{i_{0}}=\dots=x_{i_{k}}=0\}\cap V=\emptyset$. Then

[TABLE]

Apply to Lemma 1 to Lemma 4 and using the idea in [3], we get a result as follows.

Lemma 7**.**

Let $f=(f_{0}:f_{1}:\dots:f_{n}):\mathbb{C}^{m}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate meromorphic mapping on $\mathcal{P}^{1}_{c}$ with $\varsigma(f)<1.$ Let $H_{{\bf a}_{1}},\dots,H_{{\bf a}_{q}}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position which are defined by linear forms $L_{{\bf a}_{1}},\dots,L_{{\bf a}_{q}}$ respectively. Then we have the inequality

[TABLE]

in which the maximum is taken over all subsets $K$ of $\{1,\dots,q\}$ such that $\{{\bf a}_{l}\}_{l\in K}$ are linearly independent.

Proof.

Let $K\subset\{1,\dots,q\}$ be a set such that $\{{\bf a}_{l}\}_{l\in K}$ are linearly independent. Note that $q\geqslant n+1$ and the cardinal number $\#K$ of the set $K$ satisfies $\#K\leq n+1$. First of all, we consider the case $\#K=n+1.$ Let $\mathcal{T}$ be the set of all injective maps $\mu:\{0,1,\dots,n\}\rightarrow\{1,\dots,q\}$ and set $h_{l}=L_{{\bf a}_{l}}(\tilde{f})$. The quantity of left side at the inequality (2.2), say $A(r)$ simply, satisfies

[TABLE]

which further implies

[TABLE]

where

[TABLE]

By using Lemma 2, we easily find the following relation of Casorati determinants

[TABLE]

where $C_{\mu}\neq 0$ is a constant. Thus we obtain

[TABLE]

Next we estimate $B(r)$. Let $\mathbb{Z}^{+}$ be the set of positive integers and set

[TABLE]

for $t\in\mathbb{Z}^{+}$, $j\in\{0,\dots,n\}$. By a simple calculation, we find

[TABLE]

where the sum runs over all permutations

[TABLE]

of $n+1$ objects, which further implies

[TABLE]

and hence

[TABLE]

Applying Jensen’s formula to meromorphic function $h_{\mu}=\dfrac{h_{\mu(1),c}\dots h_{\mu(n),nc}}{h_{\mu(1)}\dots h_{\mu(n)}}$ on $\mathbb{C}^{m}$, we have

[TABLE]

Note that

[TABLE]

and

[TABLE]

Combining (2.5) to (2.7), we obtain

[TABLE]

Further, note that $z_{0}$ is a zero of $h_{\mu(j),jc}$ if and only if $z_{0}+jc$ is a zero of $h_{\mu(j)}.$ Thus we have

[TABLE]

for each $j=1,\dots,n.$ By using First Main Theorem

[TABLE]

and noting that

[TABLE]

then Lemma 4 deduces

[TABLE]

for each $j=1,\dots,n$, which immediately yields

[TABLE]

and combine (2.8) and (2.9), we get

[TABLE]

Combining above equality with (2.4) and using Lemma 1 , we get

[TABLE]

By using Jensen’s formula and combining (2.3) with (2), we obtain

[TABLE]

Finally, we consider the case $\#K<n+1.$ We can take $n+1-\#K$ elements from the set $\{1,\dots,q\}$ united with the set $K$ to form a new set $K^{\prime}$ such that $\{{\bf a}_{l}\}_{l\in K^{\prime}}$ are linearly independent. Note that

[TABLE]

We also obtain the estimate (2.11). The proof of Lemma 7 is completed. ∎

Lemma 8**.**

[2]** Take $c\in\mathbb{C}^{m}\setminus\{0\}$ and let $f:\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping over $\mathcal{P}_{c}^{1}$ with hyperorder $\varsigma(f)<1$. Let $H_{j}(1\leq j\leq q)$ be $q(>2N-n+1)$ hyperplanes in $N$-subgeneral position in $\mathbb{P}^{n}(\mathbb{C}).$ Then we have

[TABLE]

Lemma 9**.**

[11, 12]** Let $f:\mathbb{C}^{m}\to\mathbb{P}^{n}(\mathbb{C})$ be a linearly nondegenerate meromorphic mapping and let $\{H_{j}\}_{j=1}^{q}$ be hyperplanes in $N$-subgeneral position in $\mathbb{P}^{n}(\mathbb{C})$ with $N\geq n$ and $q>2N-n+1$. Then we have

[TABLE]

Lemma 10**.**

[14]** Take positive integers $N$ and $k$ with $N\geq k$. Let $V$ be a smooth projective subvariety of dimension $k$ in $\mathbb{P}^{n}(\mathbb{C})$ and let $D_{j}\ (1\leq j\leq N+1)$ be a hypersurface in $\mathbb{P}^{n}(\mathbb{C})$ defined by a homogenous polynomial $Q_{j}$ of same degree $d$ such that

[TABLE]

Then there exist constants $b_{tj}\in\mathbb{C}$ such that hypersurfaces $D^{*}_{t}\ (1\leq t\leq k+1)$ defined by $P_{t}$ satisfy \big{(}\cap_{t=1}^{k+1}{\rm Supp}D^{*}_{t}\big{)}\cap V=\emptyset, in which

[TABLE]

3. Proof of Theorem 1

Suppose that the ideal $\mathcal{I}(V)$ is generated by homogeneous polynomials $\mathcal{P}_{1}$, …, $\mathcal{P}_{\alpha}$ and let $\mathcal{H}_{j}\ (1\leq j\leq\alpha)$ be a hypersurface defined by $\mathcal{P}_{j}$. Let $Q_{i}\ (1\leq i\leq q)$ be a homogeneous polynomial of degree $d_{i}$ in $\mathbb{C}[x_{0},\dots,x_{n}]$ defining the hypersurface $D_{i}.$ Since we can replace $Q_{i}$ by $Q_{i}^{d/d_{i}},$ where $d$ is the least common multiple of $d_{1},\dots,d_{q}$, we may assume that $Q_{1},\dots,Q_{q}$ have the same degree $d$. By the assumption, $D_{1},\dots,D_{q}$ are in $N$-subgeneral position on $V$. Then for every subset $\mathfrak{i}=\{i_{1},\dots,i_{N+1}\}\subset\{1,\dots,q\},$ we have

[TABLE]

which means

[TABLE]

Thus by Hilbert’s Nullstellensatz [18], for each integer $\beta\in\{0,\dots,n\},$ there is an integer $m_{\beta}>\{d,\max_{t=1}^{\alpha}\{\deg\mathcal{P}_{t}\}\}$ such that

[TABLE]

where $a_{{\beta}j}$ (resp. $b_{\beta t}$) is a homogeneous polynomial of degree $m_{\beta}-d$ (resp. $m_{\beta}-\deg P_{t}$) in $\mathbb{C}[x_{0},\dots,x_{n}]$. Since $f(\mathbb{C}^{m})\subseteq V$, we have

[TABLE]

Thus we obtain an estimate

[TABLE]

where $\kappa_{\mathfrak{i}}$ is a positive constant depends only on the coefficients of $a_{{\beta}j}$ $(0\leq\beta\leq n,1\leq j\leq N+1)$. Set $A=\max_{\mathfrak{i}}\kappa_{\mathfrak{i}}$. We have

[TABLE]

Let $\mathbb{I}$ be the set of all permutations of the set $\{1,\dots,q\}$ with the cardinal number $n_{0}:=q!$, which can be listed as $\mathbb{I}=\{\mathbb{I}_{1},\dots,\mathbb{I}_{n_{0}}\},$ where $\mathbb{I}_{i}=(\mathbb{I}_{i}(1),\dots,\mathbb{I}_{i}(q))\in(\mathbb{Z}^{+})^{q}$ and $\mathbb{I}_{1}<\dots<\mathbb{I}_{n_{0}}$ according to the lexicographic order. For each $i\in\{1,\dots,n_{0}\}$, Lemm 10 implies that there exist constants $b_{itj}\in\mathbb{C}$ such that hypersurfaces $D^{*}_{i,t}\ (1\leq t\leq k+1)$ defined by $P_{i,t}$ satisfy \big{(}\cap_{t=1}^{k+1}{\rm Supp}D^{*}_{i,t}\big{)}\cap V=\emptyset, in which

[TABLE]

Then there exists a positive constant $B\geq 1$ such that

[TABLE]

for $1\leq t\leq k+1$, ${\bf x}=(x_{0},\dots,x_{n})\in\mathbb{C}^{n+1}.$ Moreover, there exists a positive constant $E$ such that

[TABLE]

Further, one defines a mapping $\widetilde{\psi}:\mathbb{C}^{n+1}\longrightarrow\mathbb{C}^{l}$ by

[TABLE]

where $l=n_{0}(k+1)$, and let ${\mathfrak{e}}=({\mathfrak{e}}_{0}:{\mathfrak{e}}_{1}:\cdots:{\mathfrak{e}}_{n}):V\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be the embedding mapping. Then a finite morphism $\psi:V\longrightarrow\mathbb{P}^{l-1}(\mathbb{C})$ is well defined by

[TABLE]

where ${\bf x}=({\mathfrak{e}}_{0}(x),...,{\mathfrak{e}}_{n}(x))$, such that $Y=\psi(V)$ is a complex projective subvarieties of $\mathbb{P}^{l-1}(\mathbb{C})$ with $\dim Y=k,$ $\deg Y:=\Delta\leq d^{k}\deg V$ (cf. [17]).

Moreover, taking a positive integer $u$ and fixed a basis $\{\phi_{0},\dots,\phi_{n_{u}}\}$ of the vector space $Y_{u}:={\mathbb{C}[y_{1},\dots,y_{l}]_{u}}/{{\mathcal{I}}_{u}(Y)}$, where $n_{u}+1=H_{Y}(u)=\dim Y_{u}$, $\mathcal{I}_{u}(Y):=\mathbb{C}[y_{1},\dots,y_{l}]_{u}\cap\mathcal{I}(Y)$ in which $\mathcal{I}(Y)$ is the prime ideal which defines algebraic variety $Y,$ according to S. D. Quang [14], a meromorphic mapping

[TABLE]

is well defined with a reduced representation $\tilde{F}=\left(\dfrac{\phi_{0}(\tilde{\psi}(\tilde{f}))}{\varphi},\dots,\dfrac{\phi_{n_{u}}(\tilde{\psi}(\tilde{f}))}{\varphi}\right),$ where $\varphi$ is a common factor of $\phi_{0}(\psi(f)),\dots,\phi_{n_{u}}(\psi(f))$ which is a holomorphic function on $\mathbb{C}^{m}$. Moreover, $F$ is linearly non-degenerate on $\mathcal{P}^{1}_{c}$. Note that

[TABLE]

Then we obtain an estimate

[TABLE]

which implies

[TABLE]

Now, we claim

[TABLE]

for $z\in\mathbb{C}^{m}\setminus\mathfrak{S}$, where $\mathfrak{S}=\mathfrak{S}_{1}\cup\mathfrak{S}_{2}$ is defined by

[TABLE]

so that (2.16) yields immediately

[TABLE]

In fact, for an fixed $z\in\mathbb{C}^{m}\setminus\mathfrak{S}$, there exists some $i\in\{1,...,n_{0}\}$ such that

[TABLE]

By (2.18) and (2.12), we have $\dfrac{||\tilde{f}(z)||^{d}}{|Q_{\mathbb{I}_{i}(j)}(\tilde{f}(z))|}\leq A$ for all $j\geq N+1$, which means

[TABLE]

It follow from (2.18) and (2.14) that

[TABLE]

and

[TABLE]

so that

[TABLE]

Since

[TABLE]

we deduce

[TABLE]

Note that $(N+1)d+(N-k)kd=N_{k}d$, where $N_{k}=(N-k+1)(k+1)$. Then (2.19) yields the claim (2.16) immediately.

Next, by using the auxiliary mapping $F:\mathbb{C}^{m}\longrightarrow\mathbb{P}^{n_{u}}(\mathbb{C})$, we claim

[TABLE]

where $N_{C(F)}(r)$ is the valence function for the zeros of Casorati determinant $C(F)$ of $F$. Fixed a point $z\in\mathbb{C}^{m}\setminus\mathfrak{S}$ and taking an index $i\in\{1,\dots,n_{0}\}$, we define

[TABLE]

where $c_{i,t,z}=\log\dfrac{||\tilde{f}(z)||^{d}||P_{i,t}||}{|P_{i,t}(\tilde{f}(z))|}$ for $i\in\{1,\dots,n_{0}\},t\in\{1,\dots,k+1\}$, and $\mathbb{R}_{+}$ is the set of nonnegative real numbers. Then according to the definition of the $u$-th Hilbert weight $S_{Y}(u,{\bf c}_{z})$ of $Y$ with respect to ${\bf c}_{z}$, there exists a subset $A_{z}\subset\{0,\dots,l_{u}\}$ with $|A_{z}|=n_{u}+1=H_{Y}(u)$ in which $l_{u}=\binom{l+u-1}{u}-1$ such that $\{{\bf x}^{{I}_{j,z}}:j\in A_{z}\}$ is a basis of the vector space $Y_{u}$ and

[TABLE]

where ${\bf x}\in\mathbb{C}^{l}$, and

[TABLE]

that is,

[TABLE]

or equivalently,

[TABLE]

Since $\{\phi_{0},\dots,\phi_{n_{u}}\}$ also is a basis of $Y_{u}$, there exist $n_{u}+1$ linear forms

[TABLE]

which are linearly independent such that

[TABLE]

Note that

[TABLE]

Then we obtain

[TABLE]

Thus we have

[TABLE]

and hence

[TABLE]

Further, by using Lemma 5 and Lemma 6, we obtain easily

[TABLE]

Note that when $z$ is changed, the subset $A_{z}$ of $\{0,...,l_{u}\}$ runs only over a finite set, so that $\mathcal{L}=\cup_{z}\mathcal{L}_{z}$ also is a finite set with $|\mathcal{L}|\leq\binom{l_{u}+1}{n_{u}+1}$. Combine (3) and (2.22), we get

[TABLE]

where $J$ is a subset of $\mathcal{L}$ with $|J|=H_{Y}(u)$ such that $\{L:L\in J\}$ is linearly independent. Applying Lemma 7 to $F$, we see

[TABLE]

Thus by (2.17) and (3), we have

[TABLE]

Hence the claim (3) follows from (2.15) and First Main Theorem immediately.

Thirdly, we claim

[TABLE]

which will follows from a functional inequality

[TABLE]

In fact, (3) is trivial over $\mathbb{C}^{m}\setminus\mathfrak{S}_{1}$. Obviously, we only need to prove (3) for $z\in\mathfrak{S}_{1}\backslash I_{f}.$ Write

[TABLE]

where $\mathfrak{T}(i)$ is the set of all points $z\in\mathfrak{S}_{1}\backslash I_{f}$ satisfying (2.18). Take $z\in\mathfrak{S}_{1}\backslash I_{f}$. W.l.o.g., we may assume $z\in\mathfrak{T}(1)$, where $\mathbb{I}_{1}=(1,\dots,q).$ Since $D_{1},\dots,D_{q}$ are in $N$-subgeneral position in $V,$ then there are at most $N$ hypersurfaces in $\{D_{1},\dots,D_{q}\}$ intersecting with $V.$ Thus (2.18) with $i=1$ implies that there exists $\mathfrak{p}\in\{1,\dots,N\}$ such that $\nu_{Q_{j}(\tilde{f})}(z)>0$ for all $j=1,\dots,\mathfrak{p},$ and $\nu_{Q_{j}(\tilde{f})}(z)=0$ for all $j=\mathfrak{p}+1,\dots,q.$ Hence by renumbering the set $\{1,\dots,q\}$ if necessary, we may assume that

[TABLE]

Thus we further claim

[TABLE]

In fact, for the case $\mathfrak{p}=1$, (2.29) follows easily from

[TABLE]

When $\mathfrak{p}\geq 2$, we next prove (2.29) by distinguishing two cases.

Case 1. $k=1.$

For this case, there is only one hypersurface in $D_{1,1}^{*},\dots,D_{1,k+1}^{*}$ intersecting with $V$, which is just $D_{1,1}^{*}$ since $\nu_{P_{1,1}(\tilde{f})}(z)=\nu_{Q_{1}(\tilde{f})}(z)>0$. Thus there are at most $N-k$ functions in $\{Q_{2}(\tilde{f}),\dots,Q_{N-k+2}(\tilde{f})\}$ vanishing at $z,$ that is, $N-k+2>\mathfrak{p}.$ Indeed, if all these functions vanish at $z,$ then $\nu_{P_{1,2}(\tilde{f})}(z)>0$, that is, $D_{1,2}^{*}$ intersects with $V$. This is a contradiction. Hence (2.29) follows from

[TABLE]

Case 2. $k\geq 2.$

Moreover, if $N-k+2>\mathfrak{p},$ then the estimate (2.31) still holds, so that it is sufficient to consider the case $N-k+2\leq\mathfrak{p}$. Thus there exists $\mathfrak{l}\in\{2,k+1\}$ such that $N-k+\mathfrak{l}=\mathfrak{p}.$ By the definition of $P_{1,t}$ at (2.13), we see easily

[TABLE]

for all $t=2,\dots,\mathfrak{l}$. Since there exists $\mathfrak{h}\in\{1,\dots,k\}$ such that $\mathfrak{h}$ hypersurfaces in $D_{1,1}^{*},\dots,D_{1,k+1}^{*}$ intersect with $V$, (2.32) means $2\leq\mathfrak{l}\leq\mathfrak{h}\leq k.$ Note that any $n_{u}$-successive and $c$-separated point $z$ of $Q_{N-k+t}(\tilde{f})\ (t=2,\dots,\mathfrak{l})$ is a $n_{u}$-successive and $c$-separated point of $P_{1,t}(\tilde{f})\ (2\leq t\leq\mathfrak{l})$, that is, $\nu_{Q_{N-k+t}(\tilde{f})}^{[n_{u},c]}(z)=\nu_{Q_{N-k+t}(\tilde{f})}(z)$ and $\nu_{P_{1,t}(\tilde{f})}^{[n_{u},c]}(z)=\nu_{P_{1,t}(\tilde{f})}(z)$ for $t=2,\dots,\mathfrak{l}.$ We see

[TABLE]

which implies

[TABLE]

Hence the claim (2.29) is proved.

According to the definition of $u$-th Hilbert weight $S_{Y}(u,{\bf c})$ with respect to

[TABLE]

with $c_{i,t}=\nu_{P_{i,t}(\tilde{f})}^{[n_{u},c]}(z)$, there exist $H_{Y}(u)$ multi-indies

[TABLE]

such that ${\bf x}^{{I}_{1}},\dots,{\bf x}^{{I}_{H_{Y}(u)}}$ is a basis of the vector space $Y_{u}$ for ${\bf x}\in\mathbb{C}^{l}$ and

[TABLE]

Moreover, there exist $H_{Y}(u)$ linear forms $L_{1},\dots,L_{H_{Y}(u)}$ which are linearly independent such that

[TABLE]

which implies

[TABLE]

so that

[TABLE]

Note that there exists a constant $C\neq 0$ such that

[TABLE]

and hence

[TABLE]

Combining (2.35) and (2.34), we obtain

[TABLE]

Since $D_{1,1}^{*}\dots,D_{1,k+1}^{*}$ are in general position in $V,$ Lemma 5 and Lemma 6 yield

[TABLE]

Thus we have

[TABLE]

It follows from (2.29) and (3) that

[TABLE]

which means

[TABLE]

since $\nu_{C(F)}(z)\geq\nu_{C(F)}^{[n_{u},c]}(z)$, $\nu_{\varphi}^{[n_{u},c]}(z)\leq\nu_{\varphi}(z)$ and $\nu_{P_{i,t}(\tilde{f})}^{[n_{u},c]}(z)\leq\nu_{P_{i,t}(\tilde{f})}(z)$. Thus the claim (3) follows from (3) and the fact

[TABLE]

Finally, the claims (3), (3) and First Main Theorem for $D_{i,t}^{*}$ yield

[TABLE]

Now, for any $\varepsilon>0,$ we choose $u=N_{k}(2k+1)\Delta lI(\varepsilon^{-1})$, so that

[TABLE]

holds. Thus it follows from (2.38) and (2.39) that

[TABLE]

Generally, replacing $D_{i}$ by $D_{i}^{d/d_{i}}$ with $d={\rm lcm}\{d_{1},\dots,d_{q}\}$, we see that

[TABLE]

Hence (2.40) implies

[TABLE]

Note that $\deg Y=\Delta\leq d^{k}\deg V,$ $\dim Y=k$ and

[TABLE]

(cf. [17]). For the choice of $u,$ we have

[TABLE]

and hence the proof of Theorem 1 is completed.

4. Proof Theorem 2

Applying Lemma 9 to the meromorphic mapping

[TABLE]

and the collection of hyperplanes $\mathfrak{H}=\{H_{1},\dots,H_{q}\}$ in $N$-subgeneral position in $\mathbb{P}^{n_{\mathcal{D}}}(\mathbb{C}),$ we have

[TABLE]

so that Theorem 2 follows from the facts $T_{G}(r)=dT_{f}(r)$ and

[TABLE]

5. Proof of Theorem 3

Assume, to the contrary, that $f\not\equiv g.$ Then there exists two distinct indices $s,t\in\{0,\dots,n\}$ such that $\Phi:=f_{s}g_{t}-f_{t}g_{s}\not\equiv 0$. The conditions ${\bf(a)}$ and ${\bf(b)}$ means that points in $\cup_{j=1}^{q}\overline{E}_{m_{j})}(D_{j},f)\backslash I_{f}$ are zeros of $\Phi$, so that

[TABLE]

Note that

[TABLE]

[TABLE]

that is

[TABLE]

Applying Theorem 2 and First Main Theorem, we have

[TABLE]

which implies

[TABLE]

where

[TABLE]

Similarly, we get

[TABLE]

Combining (2.41), (2.42) and (2.43), we have

[TABLE]

which means $q^{\prime}-\dfrac{2m_{1}n_{\mathcal{D}}}{d^{\prime}(m_{1}+1)}\leq 0$. This is contradiction. Then $f\equiv g$ and the proof of Theorem 3 is completed.

6. Proof of Theorem 4

Applying Lemma 8 to the meromorphic mapping

[TABLE]

and the collection of hyperplanes $\mathfrak{H}=\{H_{1},\dots,H_{q}\}$ in $N$-subgeneral position in $\mathbb{P}^{n_{\mathcal{D}}}(\mathbb{C}),$ we have

[TABLE]

Thus Theorem 4 follows from $T_{G}(r)=dT_{f}(r)$ and the fact

[TABLE]

7. Proof of Theorem 5

We consider the collection of hyperplanes $\mathfrak{H}=\{H_{1},\dots,H_{q}\}$ associated with $\{Q_{1}^{*},\dots,Q_{q}^{*}\}$ in $N$-subgeneral position in $\mathbb{P}^{n_{\mathcal{D}}}(\mathbb{C})$, which are defined by

[TABLE]

and use the meromorphic mapping

[TABLE]

with a reduced representation $\tilde{G}=(\tilde{f}^{I_{0}},\dots,\tilde{f}^{I_{n_{\mathcal{D}}}})$. Then

[TABLE]

satisfies $\tau({G_{j}}^{-1}(0))\subset{G_{j}}^{-1}(0)$, where each point in ${G_{j}}^{-1}(0)$ is counted multiplicity. We say that $i\sim j$ if $G_{i}=\gamma G_{j}$ for some $\gamma\in\mathcal{P}_{c}^{1}\setminus\{0\}.$ Therefore, we can split the set $\Lambda=\{1,\dots,q\}$ into disjoint equivalence classes $S_{j}$ such that $\Lambda=\cup_{j=1}^{l}S_{j}.$ First of all, assume that there exists $j\in\{1,\dots,l\}$ such that $S_{j}$ has at most $q-N-1$ elements. Put $R=\Lambda\setminus S_{j},$ then $|R|\geq N+1.$ Let $s_{0}\in S_{j}$ and put $U=R\cup\{s_{0}\}.$ Without loss of generality, we may suppose that $U=\{s_{0},s_{1},\dots,s_{N+1}\}.$ Since hyperplanes $H_{1},\dots,H_{q}$ are in $N$-subgeneral position in $\mathbb{P}^{n_{\mathcal{D}}}(\mathbb{C}),$ then exists $\gamma_{0}\in\mathbb{C}\setminus\{0\}$ and complex numbers $\gamma_{j},j=1,\dots,N+1,$ are not simultaneous with zero such that $\sum_{j=0}^{N+1}\gamma_{j}H_{s_{j}}=0.$ Hence

[TABLE]

By hypothesis of Theorem 5, we see that all zeros of $\gamma_{j}G_{j}$ are forward invariant with respect to the translation $\tau(z)=z+c.$ We obtain a meromorphic mapping

[TABLE]

with hyper-order $\varsigma(\mathcal{G})<1.$ By Lemma 3, we have $\gamma_{0}G_{s_{0}}\equiv 0,$ then $H_{s_{0}}(\tilde{G})\equiv 0.$ Thus, the image $f(\mathbb{C}^{m})$ is contained the hypersurface $D_{s_{0}}$ of $\mathbb{P}^{n}(\mathbb{C})$.

Secondly, assume that $S_{j}$ has at least $q-N$ elements for all $j=1,\dots,l.$ Then $l\leq\dfrac{q}{q-N}.$ Since $\{H_{j}\}_{j=1,\dots,q}$ is in $N$-subgeneral position, we can choose a subset $V\subset\{1,\dots,q\}$ with $|V|=n_{\mathcal{D}}+1$ such that $\{H_{j}\}_{j\in V}$ is linearly independent. Put $V_{j}=V\cap S_{j},$ then we have $V=\cup_{j=1}^{l}V_{j}.$ Since each $V_{j}$ raise to $|V_{j}|-1$ equations over the field $\mathcal{P}_{c}^{1},$ then there are at least

[TABLE]

linearly independent relations over field $\mathcal{P}_{c}^{1}.$ Hence image of $G$ is contained the projective linear subspace over $\mathcal{P}_{c}^{1}$ with dimension at most \Big{[}\dfrac{N}{q-N}\Big{]}. Specially, if $q\geq 2N+1,$ then $G(z+c)\equiv G(z),$ this implies $f(z)\equiv f(z+c).$