On Cartan's second main theorems for holomorphic curves on M-punctured complex planes

TL;DR

This paper extends key theorems in Nevanlinna-Cartan theory for holomorphic curves on M-punctured complex planes and improves results on the uniqueness problem related to inverse images of hypersurfaces.

Contribution

It provides new extensions of fundamental theorems in Nevanlinna-Cartan theory and advances the understanding of uniqueness problems for holomorphic curves.

Findings

Extended Nevanlinna-Cartan theorems for M-punctured planes

Improved results on the uniqueness of holomorphic curves

Application to inverse image problems of hypersurfaces

Abstract

In this paper, we give some extension of fundamental theorems in Nevanlinna - Cartan theory for holomorphic curve on M punctured complex planes. As an application, we establish a result for uniqueness problem of holomorphic curve by inverse image of a hypersurface, it is improvement of some results before [8, 14] in this trend.

Equations

$${\Omega }_t=D_t(0)\setminus {\cup }_{j=1}^M{\overline{D}}_{1/t}(c_j).$$

$$T_f(r)=\frac{1}{2\pi }{\int }_0^{2\pi }\log \Vert f(r{e}^{i\theta })\Vert d\theta +\frac{1}{2\pi }\sum _{j=1}^M{\int }_0^{2\pi }\log \Vert f(c_j+\frac{1}{r}{e}^{i\theta })\Vert d\theta ,$$

$$T_f(r)=\frac{1}{2\pi }{\int }_0^{2\pi }\log \Vert f(r{e}^{i\theta })\Vert d\theta +O(1).$$

$$\frac{1}{2\pi }\sum _{j=1}^M{\int }_0^{2\pi }\log \Vert f(c_j+\frac{1}{r}{e}^{i\theta })\Vert d\theta$$

$$L(z_0,\dots ,z_n)=\sum _{j=0}^n{a}_j{z}_j$$

$$(H,f)=(a,f)=\sum _{j=0}^n{a}_j{f}_j.$$

$$m_f(r,H)=\frac{1}{2\pi }{\int }_0^{2\pi }\log \frac{\Vert f(r{e}^{i\theta })\Vert }{|(a,f)(r{e}^{i\theta })|}d\theta +\frac{1}{2\pi }\sum _{j=1}^M{\int }_0^{2\pi }\log \frac{\Vert f(c_j+\frac{1}{r}{e}^{i\theta })\Vert }{|(a,f)(c_j+\frac{1}{r}{e}^{i\theta })|}d\theta ,$$

$$N_f(r,H)={\int }_{r_0}^r\frac{n_f(t,H)}{t}dt.$$

$$N_f^{\delta }(r,H)={\int }_{r_0}^r\frac{n_f^{\delta }(t,H)}{t}dt.$$

$$\bigcap _{k=1}^{n+1}\text{supp}(H_{i_k})=\varnothing ,$$

$$m_f(r,D)=\frac{1}{2\pi }{\int }_0^{2\pi }\log \frac{\Vert f(r{e}^{i\theta }){\Vert }^d}{|Q(f)(r{e}^{i\theta })|}d\theta +\frac{1}{2\pi }\sum _{j=1}^M{\int }_0^{2\pi }\log \frac{\Vert f(c_j+\frac{1}{r}{e}^{i\theta }){\Vert }^d}{|Q(f)(c_j+\frac{1}{r}{e}^{i\theta })|}d\theta ,$$

$$N_f(r,D)={\int }_{r_0}^r\frac{n_f(t,D)}{t}dt.$$

$$N_f^{\delta }(r,D)={\int }_{r_0}^r\frac{n_f^{\delta }(t,D)}{t}dt.$$

$$V\cap \text{Supp}{D}_{i_0}\cap \cdots \cap \text{Supp}{D}_{i_n}=\varnothing ,$$

$$d{T}_f(r)=m_f(r,D)+N_f(r,D)+O(1),$$

$$T_f(r)=m_f(r,H)+N_f(r,H)+O(1),$$

$$\Vert (q-n-1)T_f(r)⩽\sum _{l=1}^q{N}_f^n(r,H_l)+O(\log r+\log T_f(r)).$$

$$\alpha \ge \frac{n^n{d}^{n^2+n}(19nI({\epsilon }^{-1}){)}^n(\mathrm{deg}V{)}^{n+1}}{n!},$$

$$\Vert (q(1-\epsilon /3)-(n+1)-\epsilon /3)T_f(r)⩽\sum _{l=1}^q{d}_l^{-1}{N}_f^{\alpha }(r,Q_l)+O(\log r+\log T_f(r)).$$

$$\alpha \ge \frac{n^n{d}^{n^2+n}(19nI({\epsilon }^{-1}){)}^n(\mathrm{deg}V{)}^{n+1}}{n!},$$

$$(q(1-\epsilon /3)-(n+1)-\epsilon /3)T_f(r)⩽\sum _{l=1}^q{d}_l^{-1}{N}_f^{\alpha }(r,Q_l)+O(\log r+\log T_f(r)),$$

$$\Vert (n-(d+N+1)N)T_f(r)$$

$$\le N_f^N(r,D)+o(T_f(r)).$$

$$D=\{z\in {\mathbb{P}}^N(\mathbb{C}),\sum _{i=0}^N(\sum _{t=0}^i{x}_t{)}^n{x}_i^d=0\}$$

$$m_0(r,f)$$

$$-\frac{1}{2\pi }{\int }_0^{2\pi }{\log }^{+}|f(r_0{e}^{i\theta })|-\frac{1}{2\pi }\sum _{j=1}^M{\int }_0^{2\pi }{\log }^{+}|f(c_j+\frac{1}{r_0}{e}^{i\theta })|d\theta .$$

$$N_0(r,f)=\frac{1}{2\pi }{\int }_{r_0}^r\frac{n_0(t,f)}{t}dt.$$

$$T_0(r,f)=N_0(r,f)+m_0(r,f)$$

$$\displaystyle N_{0}\bigg{(}r,\dfrac{1}{f}\bigg{)}-N_{0}(r,f)$$

$$-\frac{1}{2\pi }{\int }_0^{2\pi }\log |f(r_0{e}^{i\theta })|d\theta -\frac{1}{2\pi }\sum _{j=1}^M{\int }_0^{2\pi }\log |f(c_j+\frac{1}{r_0}{e}^{i\theta })|d\theta .$$

$$\Vert m_0(r,\frac{f^{\prime }}{f})=O(\log r+\log T_0(r,f)),$$

$$H_X(m):=\dim \mathbb{C}[x_0,\dots ,x_N{]}_m/I_X(m).$$

Full text

2010 Mathematics Subject Classification. Primary 32H30.

Key words: Fundamental theorem, holomorphic curve, $M$ - punctured complex planes, Nevanlinna theory.

1. Introduction and main results

One of the main topic in studying the meromorphic functions is the Nevanlinna theory. In 1933, H. Cartan [2] generalized the Nevanlinna theory for holomorphic curve in projective space which is now called the Nevanlinna - Cartan theory. Since that time, this problem has been studied intensively by many authors. Nevanlinna - Cartan theory has found various applications in complex analysis and geometry complex such as uniqueness set theory, normal family theory, problem extension of holomorphic mapping and the property hyperbolic of algebraic variety.

In 2004, M. Ru [16] proved the second main theorem for hypersurface the counterpart of the result of Corvaja and Zannier [5] in approximation diophantine. In 2009, M. Ru [17] proved the second main theorem for holomorphic curves from $\mathbb{C}$ into Algebraic variety on projective space. In 2010, Z. Chen, M. Ru and Q. Yan [3] gave an improvement of Ru’s result [17] with the level of truncation. For meromorphic functions on Annuli in complex plane $\mathbb{C}$, in 2005, A. Y. Khrystiyanyn and A. A. Kondratyuk [10, 11] showed some problems for the second main theorem and defect relation. In 2007, M. O. Hanyak and A. A. Kondratyuk [9] generalized the second main theorem of A. Y. Khrystiyanyn and A. A. Kondratyuk for $M-$ punctured complex planes. In 2015, H. T. Phuong and N. V. Thin [13] have been generalized the results of A. Y. Khrystiyanyn and A. A. Kondratyuk for holomorphic curves on annuli intersecting a finite set of fixed hyperplanes in general position in $\mathbb{P}^{n}(\mathbb{C})$ with ramification. In this paper, we will prove some fundamental theorems for holomorphic mappings from $M$ - punctured complex planes to $\mathbb{P}^{n}(\mathbb{C})$ intersecting a finite set of hyperplanes which is an extension the result of Phuong and Thin [13]. Futhermore, we extend the second main theorem of M. Ru [17] for holomorphic curves from $M$ - punctured complex planes into Algebraic variety in $\mathbb{P}^{n}(\mathbb{C})$ intersecting a finite set of hypersurfaces in general position.

First we remind some definitions in [9]. Let $c_{j}\in\mathbb{C}$, $j\in\{1,\dots,M\}$ be the set of distinct points, where $M\geq 2$ is a positive integer. Then, $\Omega=\mathbb{C}\setminus\cup_{j=1}^{M}\{c_{j}\}$ is called the $M-$ punctured planes. Denotes $d=\dfrac{1}{2}\mathop{\mathrm{min}}\{|c_{j}-c_{k}|:j\neq k\}$ and $r_{0}=1/d+\max\{|c_{j}|:1\leq j\leq M\}.$ It is easy to see that $1/{r_{0}}<d,\overline{D}_{1/r_{0}}(c_{j})\cap\overline{D}_{1/r_{0}}(c_{k})=\emptyset,$ $j\neq k,$ and $\overline{D}_{1/r_{0}}(c_{j})\subset D_{r_{0}}(0)$, $j\in\{1,2,\dots,M\},$ where $D_{r}(c)$ denotes a disk of radius $r>0$ centered at $c.$ For an arbitrary $t\geq r_{0},$ we define

[TABLE]

Using the above notation, we conclude that $\Omega_{r_{0}}\subset\Omega_{r},r_{0}<r\leq+\infty.$

Let $f=(f_{0}:\dots:f_{n}):\Omega\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic map where $f_{0},\dots,f_{n}$ are holomorphic functions and without common zeros in $\Omega$. For $r_{0}<r<+\infty$, the characteristic function $T_{f}(r)$ of $f$ is defined by

[TABLE]

where $\|f(z)\|=\max\{|f_{0}(z)|,\dots,|f_{n}(z)|\}$. The above definition is independent, up to an additive constant, of the choice of the reduced representation of $f$. Futhermore, when holomorphic curve $f$ is holomorphic at $c_{j},j=1,\dots,M,$ we have

[TABLE]

Therefore, $T_{f}(r)$ is Nevanlinna-Cartan characteristic function of holomorphic curve $f$ on $\mathbb{C}.$ Thus, our definition is an extension the definition of characteristic function for holomorphic curve on $\mathbb{C}$ to $\Omega=\mathbb{C}\setminus\{c_{1},\dots,c_{M}\}.$ We add the quantity

[TABLE]

in the original definition of $T_{f}(r)$ to control the growth of f in the neighborhood of the essential singularity points.

Let $H$ be a hyperplane in $\mathbb{P}^{n}(\mathbb{C})$ and

[TABLE]

be linear form defined $H$, where $a_{j}\in\mathbb{C},\ j=0,\dots,n,$ are constants. Denote $a=(a_{0},\dots,a_{n})$ by the non-zero associated vector with $H$. And denote

[TABLE]

Under the assumption that $(a,f)\not\equiv\ 0$, for $r_{0}<r<+\infty$, the proximity function of $f$ with respect to $H$ is defined as

[TABLE]

where the above definition is independent, up to an additive constant, of the choice of the reduced representation of $f$.

We denote $n_{f}(r,H)$ by the number of zeros of $(a,f)$ in $\overline{\Omega}_{r}.$ The counting function of $f$ is defined by

[TABLE]

Now let $\delta$ be a positive integer, we denote $n_{f}^{\delta}(r,H)$ by the numbers of zeros of $(a,f)$ in $\overline{\Omega}_{r},$ where any zero of multiplicity greater than $\delta$ is “truncated” and counted as if it only had multiplicity $\delta$. The truncated counting function of $f$ is defined by

[TABLE]

Recall that hyperplanes $H_{1},\dots,H_{q},\ q>n$, in $\mathbb{P}^{n}(\mathbb{C})$ are said to be in general position if for any distinct $i_{1},\dots,i_{n+1}\in\{1,\dots,q\}$,

[TABLE]

this is equivalence to the $H_{i_{1}},\dots,H_{i_{n+1}}$ being linearly independent.

In the case of hypersurface, we may define the proximity function, counting functions of holomorphic curve $f$ similarly. Let $D$ be a hypersurface in $\mathbb{P}^{n}(\mathbb{C})$ of degree $d$. Let $Q$ be the homogeneous polynomial of degree $d$ defining $D$. Under the assumption that $Q(f)\not\equiv 0.$ Then, the proximity function $m_{f}(r,D)$ of $f$ is defined by

[TABLE]

where the above definition is independent, up to an additive constant, of the choice of the reduced representation of $f$. The next, we denote $n_{f}(r,D)$ by the number of zeros of $Q(f)$ in $\overline{\Omega}_{r}.$ The counting function $N_{f}(r,D)$ of $f$ is defined by

[TABLE]

Now let $\delta$ be a positive integer, we denote $n_{f}^{\delta}(r,D)$ by the numbers of zeros of $Q(f)$ in $\overline{\Omega}_{r},$ where any zero of multiplicity greater than $\delta$ is “truncated” and counted as if it only had multiplicity $\delta$. The truncated counting function of $f$ is defined by

[TABLE]

Let $V\subset\mathbb{P}^{N}(\mathbb{C})$ be a smooth complex projective variety of dimension $n\geq 1.$ Let $D_{1},\dots,D_{q}$ be hypersurfaces in $\mathbb{P}^{N}(\mathbb{C}),$ where $q>n.$ The hypersurfaces $D_{1},\dots,D_{q}$ are said to be in general position on V  if for every subset $\{i_{0},\dots,i_{n}\}\subset\{1,\dots,q\},$ we have

[TABLE]

where $\text{Supp}(D)$ means the support of the divisor $D.$ A map $f:\Omega\to V$ is said to be algebraically nondegenerate if the image of $f$ is not contained in any proper subvarieties of $V.$

In this paper, a notation $``\|"$ in the inequality means that the inequality holds for $r\in(r_{0},+\infty)$ outside a set of finite measure.

Our main results are

Theorem 1**.**

Let $D$ be a hypersurfaces in $\mathbb{P}^{n}(\mathbb{C})$ defining the homogeneous polynomial $Q$ with degree $d$ and $f=(f_{0}:\dots:f_{n}):\Omega\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained $D$. Then we have for any $r_{0}<r<+\infty$,

[TABLE]

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

Note that, when $d=1$, we get the corollary as following:

Corollary 1**.**

Let $H$ be a hyperplane in $\mathbb{P}^{n}(\mathbb{C})$ and $f=(f_{0}:\dots:f_{n}):\Omega\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained $H$. Then we have for any $r_{0}<r<+\infty$,

[TABLE]

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

Theorem 2**.**

Let $f=(f_{0}:\dots:f_{n}):\Omega\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Then we have for any $r_{0}<r<+\infty,$

[TABLE]

Theorem 3**.**

Let $V\subset\mathbb{P}^{N}(\mathbb{C})$ be a complex projective variety of dimension $n\geq 1.$ Let $D_{1},\dots,D_{q}$ be hypersurfaces in $\mathbb{P}^{N}(\mathbb{C})$ of degree $d_{j},$ located in general position on $V.$ Let $d$ be the least common multiple of the $d_{i},$ $i=1,\dots,q.$ Let $f=(f_{0}:\dots:f_{N}):\Omega\rightarrow V$ be an algebraically non-degenerate holomorphic map. Let $\varepsilon>0$ and

[TABLE]

where $I(x):=\mathop{\mathrm{min}}\{k\in\mathbb{N}:k>x\}$ for a positive real number $x.$ Then

[TABLE]

When $f$ is holomorphic at $c_{j},j=1,\dots,M,$ we get following result of Ru et. al [3]. Thus Theorem 3 is an extension of Ru et. al [3] and Ru [17] for holomorphic curve with essential singularity points.

Theorem 4**.**

Let $V\subset\mathbb{P}^{N}(\mathbb{C})$ be a complex projective variety of dimension $n\geq 1.$ Let $D_{1},\dots,D_{q}$ be hypersurfaces in $\mathbb{P}^{N}(\mathbb{C})$ of degree $d_{j},$ located in general position on $V.$ Let $d$ be the least common multiple of the $d_{i},$ $i=1,\dots,q.$ Let $f=(f_{0}:\dots:f_{N}):\mathbb{C}\rightarrow V$ be an algebraically non-degenerate holomorphic map. Let $\varepsilon>0$ and

[TABLE]

where $I(x):=\mathop{\mathrm{min}}\{k\in\mathbb{N}:k>x\}$ for a positive real number $x.$ Then

[TABLE]

holds for all $r\in(0,+\infty)$ outside a set of finite measure.

Theorem 1 and Corollary 1 is first main theorem, and Theorem 2 is second main theorem for holomorphic curves from $M-$ punctured $\Omega$ to $\mathbb{P}^{n}(\mathbb{C})$ intersecting a collection of fixed hyperplanes in general position with truncated counting functions. Theorem 3 is second main theorem for holomorphic curves from $M-$ punctured $\Omega$ to $V$ intersecting a collection of fixed hypersurfaces in general position with the level of truncation. When one applies inequalities of second main theorem type, it is often crucial to the application to have the inequality with truncated counting functions. For example, all existing constructions of unique range sets depend on a second main theorem with truncated counting functions.

Theorem 5**.**

Let $f:\Omega\to\mathbb{P}^{N}(\mathbb{C})$ be an algebraically nondegenerate holomorphic curve. Let $d$ and $n$ be two integers with $n>N(d+N+1).$ Let $\mathcal{H}_{i}=\{z\in\mathbb{P}^{N}(\mathbb{C}),\mathcal{H}_{i}(z)=0\},0\leq i\leq N,$ be hyperplanes in $\mathbb{P}^{N}(\mathbb{C}).$ Let $D_{i}=\{z\in\mathbb{P}^{N}(\mathbb{C}),Q_{i}(z)=0\},0\leq i\leq N,$ be hypersurfaces of degree $d$ such that the hypersurfaces $\{\mathcal{H}_{0}^{n}Q_{0}=0\},\dots,\{\mathcal{H}_{N}^{n}Q_{N}=0\}$ are in general position in $\mathbb{P}^{N}(\mathbb{C}).$ Let $D=\{z\in\mathbb{P}^{N}(\mathbb{C}),\sum_{i=0}^{N}\mathcal{H}_{i}^{n}Q_{i}=0\}.$ Then

[TABLE]

We give a hypersurfaces satisfying Theorem 5.

Example 6**.**

Let $D_{i}=\{z=(x_{0}:\dots:x_{N})\in\mathbb{P}^{N}(\mathbb{C}),x_{i}^{d}=0\},0\leq i\leq N,$ be hypersurfaces of degree $d.$ Let $\mathcal{H}_{i}=\{z=(x_{0}:\dots:x_{N})\in\mathbb{P}^{N}(\mathbb{C}),\sum_{t=0}^{i}x_{t}=0\}.$ We see that the hypersurfaces $\{(\sum_{t=0}^{i}x_{t})^{n}x_{i}^{d}=0\},0\leq i\leq N,$ are in general position in $\mathbb{P}^{N}(\mathbb{C}).$ Then

[TABLE]

satisfies the Theorem 5 with $n>N(d+N+1).$

As an application of Theorem 5, we prove the uniqueness theorem for holomorphic curves on $M-$ punctured complex plane by inverse image of a Fermat hypersurface.

Theorem 7**.**

Let $f,g:\Omega\to\mathbb{P}^{N}(\mathbb{C})$ be two algebraically nondegenerate holomorphic curves, and $n$ be a integer with $n>N(d+N+3).$ Let $D$ be a hypersurface as in Theorem 5. Suppose that $f(z)=g(z)$ on $f^{-1}(D)\cup g^{-1}(D),$ then $f\equiv g.$

When $f$ and $g$ are holomorphic at $c_{j},j=1,\dots,M,$ we obtain the corollary as following:

Corollary 2**.**

Let $f,g:\mathbb{C}\to\mathbb{P}^{N}(\mathbb{C})$ be two algebraically nondegenerate holomorphic curves, and $n$ be a integer with $n>N(d+N+3).$ Let $D$ be a hypersurface as in Theorem 5. Suppose that $f(z)=g(z)$ on $f^{-1}(D)\cup g^{-1}(D),$ then $f\equiv g.$

We reduce the number hypersurfaces in before results. The authors [6, 8, 14] studied the uniqueness problem with a number lager hypersurfaces. Here, we only need a hypersurface.

2. Some preliminaries in Nevanlinna theory for meromorphic functions

In order to prove theorems, we need the following lemmas. Let $f$ be a meromorphic function on $M-$ punctured $\Omega$ and $r\in(r_{0},+\infty)$, we recall that

[TABLE]

We denote $n_{0}(r,f)$ by the numbers of its poles in $\overline{\Omega}_{r}.$ The counting function $N_{0}(r,f)$ of $f$ is defined by

[TABLE]

The function

[TABLE]

is called the Nevanlinna characteristic of $f$.

Lemma 1**.**

[9]** (Jensen’s Theorem for $M-$ punctured planes) Let $f$ be a non-constant meromorphic function in an $M-$ punctured plane $\Omega$ not identically equal to zero and let $r_{0}<r<+\infty$. Then

[TABLE]

Lemma 2**.**

[9]** Let $f$ be a non-constant meromorphic function on $\Omega.$ Then, we have the equality

[TABLE]

holds for all $r\in(r_{0},+\infty)$ outside a set of finite measure.

Let $X\subset\mathbb{P}^{N}(\mathbb{C})$ be a projective variety of dimension $n$ and degree $\Delta.$ Let $I_{X}$ be the prime ideal in $\mathbb{C}[x_{0},\dots,x_{N}]$ defining $X$. Denote by $\mathbb{C}[x_{0},\dots,x_{N}]_{m}$ the vector space of homogeneous polynomials in $\mathbb{C}[x_{0},\dots,x_{N}]$ of degree $m$ (including 0). Put $I_{X}(m):=\mathbb{C}[x_{0},\dots,x_{N}]_{m}\cap I_{X}.$ The Hilbert function $H_{X}$ of $X$ is defined by

[TABLE]

For each tuple $c=(c_{0},\dots,c_{N})\in\mathbb{R}^{N+1}_{\geq 0}$ and $m\in\mathbb{N}$, we define the $m$-th Hilbert weight $S_{X}(m,c)$ of $X$ with respect to $c$ by

[TABLE]

where $I_{i}=(I_{i0},\dots,I_{iN})\in\mathbb{N}_{0}^{N+1}$ and the maximum is taken over all sets $\{x^{I_{i}}=x_{0}^{T_{i0}}\dots x_{N}^{I_{iN}}\}$ whose residue classes modulo $I_{X}(m)$ form a basis of the vector space $C[x_{0},\dots,x_{N}]_{m}/I_{X}(m).$

Lemma 3**.**

[17]** Let $X\subset\mathbb{P}^{N}(\mathbb{C})$ be an algebraic variety of dimension $n$ and degree $\Delta$. Let $m>\Delta$ be an integer and let $c=(c_{0},\dots,c_{N})\in\mathbb{R}_{\geq 0}^{N+1}.$ Then

[TABLE]

Lemma 4**.**

[17]** Let $Y$ be a subvariety of $\mathbb{P}^{q-1}(\mathbb{C})$ of dimension $n$ and degree $\Delta.$ Let $c=(c_{1},\dots,c_{q})$ be a tupe of positive reals. Let $\{i_{0},\dots,i_{n}\}$ be a subset of $\{1,\dots,q\}$ such that $\{y_{i_{0}}=\dots=y_{i_{n}}=0\}\cap Y=\emptyset$. Then

[TABLE]

3. Proof of Theorems

Proof of Theorem 1.

The first, we note that $N_{0}(r,Q(f))=0.$ By the definitions of $T_{f}(r)$, $N_{f}(r,H)$, $m_{f}(r,H)$ and apply to Lemma 1 for $Q(f)\not\equiv 0$, we have

[TABLE]

Hence, we get

[TABLE]

This is conclusion of Theorem 1. ∎

Proof of Theorem 2.

To prove the Theorem 2, we need some lemmas. First we recall the property of Wronskian. Let $f=(f_{0}:\dots:f_{n}):\Omega\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be holomorphic curves, the determinant of Wronskian of $f$ is defined by

[TABLE]

We denote by $N_{W}(r,0)$ the counting function of zeros of $W(f_{0},\dots,f_{n})$ in $\overline{\Omega}_{r}$, namely

[TABLE]

Let $L_{0},\dots,L_{n}$ are linearly independent forms of $z_{0},\dots,z_{n}$. For $j=0,\dots,n$, set

[TABLE]

By the property of Wronskian there exists a constant $C\neq 0$ such that

[TABLE]

Lemma 5**.**

Let $f=(f_{0}:\dots:f_{n}):\Omega\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Then we have the inequality

[TABLE]

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

First, we prove

[TABLE]

holds for any $r\in(r_{0},+\infty)$. Let $K\subset\{1,\dots,q\}$ such that $a_{l},l\in K,$ are linearly independent. Without loss of generality, we may assume that $q\geqslant n+1$ and $\#K=n+1.$ Let $\mathcal{T}$ is the set of all injective maps $\mu:\{0,1,\dots,n\}\rightarrow\{1,\dots,q\}$. Then we have

[TABLE]

By the property of Wronskian, we see that

[TABLE]

where $C\neq 0$ is a constant. So we obtain

[TABLE]

Take

[TABLE]

From property of Wronskian (see [12], Proposition 1.4.3), we have

[TABLE]

We see

[TABLE]

By Lemma 2, we have

[TABLE]

From the definition of $T_{0}(r,g_{\mu(j)}^{\prime})$, $N_{0}(r,g_{\mu(j)}^{\prime})$ and (3.9), we obtain

[TABLE]

Similarly, again using Lemma 2 and (3.10), we have

[TABLE]

By argument as (3.11) and using inductive method, we obtain that the inequality

[TABLE]

holds for all $\nu\in\mathbb{N}^{*}.$ Furthemore, by Lemma 2, we also have the equality

[TABLE]

holds for all $\nu\in\mathbb{N}.$ Combining (3), (3.12) and (3.13), we get for any $k\in\{1,\dots,n\}$ and $j\in\{1,\dots,n\}$,

[TABLE]

By the definition of $T_{0}(r,g_{\mu(j)}),T_{f}(r)$, we have

[TABLE]

Hence for any $\mu\in\mathcal{T}$, from (3), (3.14) and (3), we have

[TABLE]

This implies

[TABLE]

We may obtain the inequality (3.1) from (3.2) and (3.16). Similarly, we get

[TABLE]

holds for any $r\in(r_{0},+\infty)$ and for all $j=1,\dots,M$. Combining (3.1) and (3.17) we obtain

[TABLE]

Since

[TABLE]

we have the conclusion of this lemma. ∎

Lemma 6**.**

Let $f=(f_{0}:\dots:f_{n}):\Omega\longrightarrow\mathbb{P}^{n}(\mathbb{C})$ be a linearly non-degenerate holomorphic curve and $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Let $a_{j}$ be the vector associated with $H_{j}$ for $j=1,\dots,q$. Then

[TABLE]

Proof.

Let $a_{l}=(a^{(l)}_{0},\dots,a^{(l)}_{n})$ be the associated vector of $H_{l}$, $1\leqslant l\leqslant q,$ and let $\mathcal{T}$ be the set of all injective maps $\mu:\{0,1,\dots,n\}\longrightarrow\{1,\dots,q\}$. By hypothesis $H_{1},\dots,H_{q}$ are in general position for any $\mu\in\mathcal{T},$ then the vectors $a_{\mu(0)},\dots,a_{\mu(n)}$ are linearly independent.

Let $\mu\in T$, we have

[TABLE]

Solve the system of linear equations (3.18), we get

[TABLE]

where\biggl{(}b_{j}^{\mu(t)}\biggl{)}_{t,j=0}^{n} is the inverse matrix of \biggl{(}a_{j}^{\mu(t)}\biggl{)}_{t,j=0}^{n}. So there is a constant $C_{\mu}$ satisfying

[TABLE]

Set $C=\max\limits_{\mu\in\mathcal{T}}C_{\mu}$. Then for any $\mu\in\mathcal{T}$, we have

[TABLE]

For any $z\in\Omega$, there exists the mapping $\mu\in\mathcal{T}$ such that

[TABLE]

for $l\notin\{\mu(0),\dots,\mu(n)\}$. Hence

[TABLE]

We have

[TABLE]

This implies

[TABLE]

So we obtain

[TABLE]

This is conclusion of the Lemma 3. ∎

Proof of Theorem 2.

By Lemma 5 and Lemma 6, we obtain

[TABLE]

By Theorem 1, we get that

[TABLE]

for any $j\in\{1,\dots,q\}$. So from (3.19), we have

[TABLE]

For $z_{0}\in\overline{\Omega}_{r}$, we may assume that $(a_{l},f)$ vanishes at $z_{0}$ for $1\leqslant l\leqslant q_{1}$, $(a_{l},f)$ does not vanish at $z_{0}$ for $l>q_{1}$. Hence, there exists a integer $k_{l}$ and nowhere vanishing holomorphic function $g_{l}$ in neighborhood $U$ of $z$ such that

[TABLE]

here $k_{l}=0$ for $q_{1}<l\leqslant q$. We may assume that $k_{l}\geqslant n$ for $1\leqslant l\leqslant q_{0}$, and $1\leqslant k_{l}<n$ for $q_{0}<l\leqslant q_{1}.$ By property of the Wronskian, we have

[TABLE]

where $h(z)$ is holomorphic function on $U$. Then $W(f)$ is vanishes at $z_{0}$ with order at least

[TABLE]

By the definition of $N_{f}(r,H),\ N_{W}(r,0)$ and $N_{f}^{n}(r,H)$, we have

[TABLE]

So from (3.20), we have

[TABLE]

The proof of Theorem 2 is completed. ∎

Proof of Theorem 3.

Let $D_{1},\dots,D_{q}$ be the hypersurfaces in $\mathbb{P}^{N}(\mathbb{C})$ which are located in general position on $V.$ Let $Q_{l},1\leq l\leq q$ be the homogeneous polynomials in $\mathbb{C}[x_{0},\dots,x_{N}]$ of degree $d_{l}$ defining on $D_{l}.$ We can replace $Q_{l}$ by $Q_{l}^{d/d_{l}},$ where $d$ is the $l.c.m$ of $d_{l},$ $l=1,\dots,q,$ we may assume that $Q_{1},\dots,Q_{q}$ have the same degree of $d.$

Given $z\in\Omega$ there exists a renumbering $\{i_{0},\dots,i_{n}\}$ of the indices $\{1,\dots,q\}$ such that

[TABLE]

Suppose that $P_{1},\dots,P_{s}$ is a base of algebraic variety $V.$ From the hypothesis, $D_{1},\dots,D_{q}$ are hypersurfaces in $\mathbb{P}^{N}(\mathbb{C})$ which are located in general position on $V,$ we have for every subset $\{i_{0},\dots,i_{n}\}\subset\{1,\dots,q\},$

[TABLE]

This implies

[TABLE]

Thus by Hilbert’s Nullstellensatz [19], for any integer k, $0\leq k\leq N,$ there is an integer $m_{k}>\{d,\max_{t=1}^{s}\{\deg P_{t}\}\}$ such that

[TABLE]

where $b_{k_{j}}$ are homogeneous forms with coefficients in $\mathbb{C}$ of degree $m_{k}-d$ and $b_{t}$ are homogeneous forms with coefficients in $\mathbb{C}$ of degree $m_{k}-\deg P_{t},$ $t=1,\dots,s.$ So from $f:\Omega\to V,$ we have

[TABLE]

This implies

[TABLE]

where $c_{1}$ is a positive constant depends only on the coefficients of $b_{k_{j}},0\leq j\leq n,0\leq k\leq N,$ thus depends only on the coefficients of $Q_{l},1\leq l\leq q.$ Therefore,

[TABLE]

By (3.21) and (3.36), we get

[TABLE]

where $C=c_{1}^{q-n-1}\prod_{l\not\in\{i_{0},\dots,i_{n}\}}||Q_{l}||$ and $||Q_{l}||$ is the maximum of the absolute values of the coefficients of $Q_{l}.$ Thus, we have

[TABLE]

Hence, we get

[TABLE]

By argument as M. Ru [17], we consider the map

[TABLE]

Put $Y=\psi(V).$ The hypothesis in general position implies that $\psi$ is a finite morphimsm on $V$ and $Y$ is a complex projective subvarieties of $\mathbb{P}^{q-1}(\mathbb{C})$ and $\dim Y=n,$ $\deg Y:=\Delta\leq d^{n}\deg V.$ For each $a=(a_{1},\dots,a_{q})\in\mathbb{Z}_{\geq 0}^{q},$ denote by $y^{a}=y_{1}^{a_{1}}\dots y_{q}^{a_{q}}.$ Let $m$ be a positive integer, we consider the vector space $V_{m}=\mathbb{C}[y_{1},\dots,y_{q}]_{m}/(I_{Y})_{m},$ where $I_{Y}$ is the prime ideal which is definied algebraic variety $Y,$ $(I_{Y})_{m}:=\mathbb{C}[y_{1},\dots,y_{q}]_{m}\cap I_{Y}.$ Fix a basis $\{\phi_{0},\dots,\phi_{n_{m}}\}$ of $V_{m}$, where $n_{m}+1=H_{Y}(m)=\dim V_{m}.$ Set,

[TABLE]

Note that, $f$ is algebraically non-degenerate, then $F$ is also. For any $c\in\mathbb{R}_{\geq 0}^{q}$, the Hilbert function of $Y$ with respect to the weight $c$ is definied by

[TABLE]

where the maximum is taken over all sets of monomials $y^{a_{1}},\dots,y^{a_{H_{Y}(m)}}$ whose $y^{a_{1}}+(I_{Y})_{m},\dots,y^{a_{H_{Y}(m)}}+(I_{Y})_{m}$ is a basis of $\mathbb{C}[y_{1},\dots,y_{q}]_{m}/(I_{Y})_{m}.$ For every $z\in\Omega,$ denote $c_{z}=(c_{1,z},\dots,c_{q,z})$, where $c_{l,z}=\log\dfrac{||f(z)||^{d}||Q_{l}||}{|Q_{l}(f(z))|},$ $l=1,\dots,q.$ We see that $c_{z}\in\mathbb{R}_{\geq 0}^{q},$ for all $z\in\Omega.$ There exists a subset $I_{z}\subset\{0,\dots,q_{m}\},q_{m}=C_{q+m-1}^{m}-1,|I_{z}|=n_{m}+1=H_{Y}(m)$ which $\{y^{a_{i}}:i\in I_{z}\}$ is a basis of $\mathbb{C}[y_{1},\dots,y_{q}]_{m}/(I_{Y})_{m}$ (residue classes modulo $(I_{Y})_{m}$) and

[TABLE]

From two basis $\{y^{a_{i}}:i\in I_{z}\}$ and $\{\phi_{0},\dots,\phi_{n_{m}}\},$ there exist the forms independent linearly $\{L_{l,z},l\in I_{z}\}$ such that

[TABLE]

We denote $J$ by the set of indices of the linear forms $L_{l,z}.$ We see

[TABLE]

This implies

[TABLE]

By Lemma 3, we have

[TABLE]

From Lemma 4 and $D_{1},\dots,D_{q}$ are in general position on $V$, for any $\{i_{0},\dots,i_{n}\}\subset\{1,\dots,q\}$, we have

[TABLE]

Using the definition of $c_{z}$, we obtain

[TABLE]

From (3) to (3.27), we have

[TABLE]

Take the integration of both sides of (3), we have

[TABLE]

This implies

[TABLE]

Next, apply to Lemma 5 for $F$ and collection of hyperplanes $L_{l},l\in J,$ for every $\varepsilon>0$ and $m$ is large enough, we obtain

[TABLE]

where $N_{W}(r,0)$ is denoted by the Wronskian of $F.$ Combining (3), (3) and (3), we get

[TABLE]

Using the Theorem 1, we see $T_{F}(r)\leq dmT_{f}(r)+O(1).$ Thus, (3) implies

[TABLE]

By an argument method in [3], we conclude

[TABLE]

Combining (3) and (3), we have

[TABLE]

We choose the $m$ sufficiently large such that

[TABLE]

We may choose $m=18n^{2}\Delta I(\varepsilon^{-1})$ for the inequality (3.35), where $I(x):=\mathop{\mathrm{min}}\{k\in\mathbb{N}:k>x\}$ for each positive constant $x.$ Thus, from (3) and (3.35), we get the inequality

[TABLE]

By property $\deg Y=\Delta\leq d^{n}\deg V,$ where $d=lcm\{d_{1},\dots,d_{q}\}$, $\deg Y=n$ and $n_{m}\leq\Delta C_{m+n}^{n},$ we have

[TABLE]

For the choice of $m,$ we have

[TABLE]

∎

Proof of Theorem 5.

Let ${\large\text{f}}=(f_{0}:\dots:f_{N})$ be a reduced representation of $f,$ where $f_{0},\dots,f_{N}$ are entire functions on $\Omega$ and have no common zeros. We consider the function $\phi_{i}=Q_{i}\circ{\large\text{f}}=Q_{i}(f_{0},\dots,f_{N}),0\leq i\leq N.$ Let $F=(\phi_{0}f_{0}^{n}:\dots:\phi_{N}f_{N}^{n}).$ Since the hypersurfaces $\{\mathcal{H}_{i}^{n}Q_{i}=0\},0\leq i\leq N,$ are located in general position in $\mathbb{P}^{N}(\mathbb{C}),$ then $F:\Omega\to\mathbb{P}^{N}(\mathbb{C})$ is a holomorphic curve. Let $\mathfrak{H}_{i},0\leq i\leq N,$ be the hypersurface defined by $\{\mathcal{H}_{i}^{n}Q_{i}=0\},0\leq i\leq N.$ From the hypothesis $\mathfrak{H}_{0},\dots,\mathfrak{H}_{N}$ are in general position, i.e.

[TABLE]

Thus by Hilbert’s Nullstellensatz [19], for any integer k, $0\leq k\leq N,$ there is an integer $m_{k}>n+d$ such that

[TABLE]

where $b_{0},\dots,b_{N}$ are homogeneous forms with coefficients in $\mathbb{C}$ of degree $m_{k}-(n+d).$ This implies

[TABLE]

where $c_{1}$ is a positive constant depending only on the coefficients of $b_{i},0\leq i\leq N,0\leq k\leq N,$ thus depending only on the coefficients of $Q_{i},0\leq i\leq N.$ Therefore,

[TABLE]

From (3.36) and the First Main Theorem, we have

[TABLE]

On the other hand, by applying Theorem 2 to $F$, and the hyperplanes

[TABLE]

and

[TABLE]

yields

[TABLE]

We have

[TABLE]

for all $i=0,\dots,N,$ where $N^{N}(r,\dfrac{1}{g})$ is counting function with level of truncation $N$ of $g.$ Hence

[TABLE]

for all $i=0,\dots,N.$ Also note $N_{F}^{N}(r,H_{N+1})=N_{f}^{N}(r,D).$ By combining (3) to (3), we obtain

[TABLE]

∎

Proof of Theorem 7.

We suppose that $f\not\equiv g,$ then there are two numbers $\alpha,\beta\in\{0,\dots,N\},$ $\alpha\neq\beta$ such that $f_{\alpha}g_{\beta}\not\equiv f_{\beta}g_{\alpha}.$ Assume that $z_{0}\in\Omega$ is a zero of $P(f),$ from condition $f(z)=g(z)$ when $z\in f^{-1}(D)\cup g^{-1}(D),$ we get $f(z_{0})=g(z_{0}).$ This implies $z_{0}$ is a zero of $\dfrac{f_{\alpha}}{f_{\beta}}-\dfrac{g_{\alpha}}{g_{\beta}}.$ Therefore, we have

[TABLE]

Apply to Theorem 5, we obtain

[TABLE]

Similarly, we have

[TABLE]

Combining (3.40) and (3.41), we get

[TABLE]

This is a contradiction with $n>(d+N+3)N.$ Hence $f\equiv g.$ ∎