Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

TL;DR

This paper develops new bounds for the common zeros of meromorphic functions on algebraic tori, integrating Nevanlinna theory with Diophantine approximation techniques to advance the understanding of greatest common divisors in complex analysis.

Contribution

It introduces a general form of the conjectural asymptotic gcd inequality and extends results to moving targets, combining Nevanlinna theory with Diophantine methods.

Findings

Proved a general asymptotic gcd inequality for meromorphic functions.

Extended results to moving targets in algebraic tori.

Enhanced bounds using stronger inequalities from Nevanlinna theory.

Abstract

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural "asymptotic gcd" inequality of Pasten and the second author, and consider moving targets versions of our results.

Equations

$$N_f(Y,r){\le }_{\mathrm{exc}}ϵT_f(r).$$

$$T_{\overline{Y},f}(r){\le }_{\mathrm{exc}}ϵT_f(r)$$

$$N_{\mathbf{g}}(Y,r){\le }_{\mathrm{exc}}ϵT_{\mathbf{g}}(r).$$

$$T_{\overline{Y},\mathbf{g}}(r){\le }_{\mathrm{exc}}ϵT_{\mathbf{g}}(r).$$

$$N_{\gcd }(F(g_1,\dots ,g_n),G(g_1,\dots ,g_n),r){\le }_{\mathrm{exc}}ϵ\underset{1\le i\le n}{\max }\{T_{g_i}(r)\}.$$

$$T_{\gcd }(F(g_1,\dots ,g_n),G(g_1,\dots ,g_n),r){\le }_{\mathrm{exc}}ϵ\underset{1\le i\le n}{\max }\{T_{g_i}(r)\}.$$

$$N_{\gcd }(f-1,g-1,r){\le }_{\mathrm{exc}}ϵ\max \{T_f(r),T_g(r)\}.$$

$$N_{\gcd }$$

$$+\frac{1}{1+ϵ/4}(N_f(0,r)+N_g(0,r)+N_f(\infty ,r)+N_g(\infty ,r)).$$

$$N_{\gcd }(f^n-1,g^n-1,r){\le }_{\mathrm{exc}}ϵ\max \{T_{f^n}(r),T_{g^n}(r)\}.$$

$$N_{\gcd }(F(g_1^k,\dots ,g_n^k),G(g_1^k,\dots ,g_n^k),r){\le }_{\mathrm{exc}}ϵ\underset{1\le i\le n}{\max }\{T_{g_i^k}(r)\};$$

$$T_{\gcd }(F(g_1^k,\dots ,g_n^k),G(g_1^k,\dots ,g_n^k),r){\le }_{\mathrm{exc}}ϵ\underset{1\le i\le n}{\max }\{T_{g_i^k}(r)\},$$

$$N_{\gcd }(f^k-1,g^k-1,r){\le }_{\mathrm{exc}}ϵ\max \{T_{f^k}(r),T_{g^k}(r)\}.$$

$$N_{\gcd }(F(g_1^k,\dots ,g_n^k),G(g_1^k,\dots ,g_n^k),r)=(\mathrm{deg}\gcd (F(g_1^k,\dots ,g_n^k),G(g_1^k,\dots ,g_n^k)))\log r+O(1)$$

$$\underset{1\le i\le n}{\max }\{T_{g_i^k}(r)\}=k\underset{1\le i\le n}{\max }\{\mathrm{deg}{g}_i\}\log r+O(1),$$

$$\mathrm{deg}\gcd (F(g_1^k,\dots ,g_n^k),G(g_1^k,\dots ,g_n^k))<ϵk.$$

$$G=a_{\mathbf{i}(0)}+\sum _{j=1}^{\ell }{a}_{\mathbf{i}(j)}{\mathbf{x}}^{\mathbf{i}(j)},$$

$${\mathfrak{g}}_G:=(1,{\mathbf{g}}^{\mathbf{i}(1)},\dots ,{\mathbf{g}}^{\mathbf{i}(\ell )}):\mathbb{C}\to {\mathbb{P}}^{\ell }.$$

$$T_{G(g_1,\dots ,g_n)}(r)=T_{1+a_1{g}_1+\cdots +a_n{g}_n}(r)=T_{(1,g_1,\dots ,g_n)}(r)+O(1)$$

$$\max \{T_{g_1}(r),\dots ,T_{g_n}(r)\}\le T_{(1,g_1,\dots ,g_n)}(r)\le n\max \{T_{g_1}(r),\dots ,T_{g_n}(r)\}.$$

$$\log \gcd (a^n-1,b^n-1)\le ϵn$$

$$\log \gcd (\alpha ,\beta )$$

$$N_{\gcd ,S}(\alpha ,\beta )$$

$$N_{\gcd ,S}(F(u_1,\dots ,u_n),G(u_1,\dots ,u_n))<ϵ\max \{h(u_1),h(u_2),\dots ,h(u_n)\}$$

$$\log \gcd (F(u_1,\dots ,u_n),G(u_1,\dots ,u_n))<ϵ\max \{h(u_1),h(u_2),\dots ,h(u_n)\}$$

$$N_{Y,S}(P)\le ϵh_A(P)+O(1)$$

$$h_Y(P)\le ϵh_A(P)+O(1)$$

$$v_z^{+}(f):=\max \{0,v_z(f)\},\text{and }{v}_z^{-}(f):=-\min \{0,v_z(f)\}.$$

$$N_f(\infty ,r)$$

$$=\sum _{0<|z|\le r}{v}_z^{-}(f)\log |\frac{r}{z}|+v_0^{-}(f)\log r,$$

$$N_f^{(Q)}(\infty ,r)$$

$$=\sum _{0<|z|\le r}\min \{Q,v_z^{-}(f)\}\log |\frac{r}{z}|+\min \{Q,v_0^{-}(f)\}\log r.$$

Full text

The first author was supported in part by NSF grant DMS-1352407.

The second author was supported in part by Taiwan’s MoST grant 106-2115-M-001-001-MY2.

2010 Mathematics Subject Classification: Primary 32H30; Secondary 11J97

1. Introduction

We prove upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. A starting point for such results (in a geometric formulation) comes from the study of holomorphic curves in semi-abelian varieties by Noguchi, Winkelmann, and Yamanoi, who proved the following:

Theorem 1.1** (Noguchi, Winkelmann, Yamanoi [12, Th. 5.1] (see also [11, §6.5])).**

Let $f:\mathbb{C}\to A$ be a holomorphic map to a semi-abelian variety $A$ with Zariski-dense image. Let $Y$ be a closed subscheme of $A$ with $\operatorname{codim}Y\geq 2$ and let $\epsilon>0$.

1. (a)

Then

[TABLE] 2. (b)

There exists a compactification $\overline{A}$ of $A$, independent of $\epsilon$, such that for the Zariski closure $\overline{Y}$ of $Y$ in $\overline{A}$,

[TABLE]

Here $N_{f}(Y,r)$ is a counting function associated to $f$ and $Y$, $T_{\overline{Y},f}(r)$ is a Nevanlinna characteristic (or height) function associated to $f$ and $\overline{Y}$, and $T_{f}(r)$ is any characteristic function associated to an appropriate ample line bundle (see [12] for more discussion and Section 2 for the relevant definitions from Nevanlinna theory). The notation $\leq_{\rm exc}$ means that the estimate holds for all $r$ outside a set of finite Lebesgue measure, possibly depending on $\epsilon$.

More generally, Noguchi, Winkelmann, and Yamanoi proved a result for $k$-jet lifts of holomorphic maps to semi-abelian varieties. The case when $A$ is an abelian variety was proved by Yamanoi [24].

A first goal of our work is to obtain a new proof of Theorem 1.1 when $A=(\mathbb{C}^{*})^{n}$ is the complex algebraic torus. In this case we obtain the following refinement of Theorem 1.1.

Theorem 1.2**.**

Let $Y$ be a closed subscheme of $(\mathbb{C}^{*})^{n}$ of codimension at least $2$. Let ${\bf g}=(g_{1},\cdots,g_{n})$ be a holomorphic map from $\mathbb{C}$ to $(\mathbb{C}^{*})^{n}$ with Zariski dense image (equivalently, $g_{1},\dots,g_{n}$ are entire functions without zeros, and $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin\mathbb{C}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$). Let $\epsilon>0$.

1. (a)

Then

[TABLE] 2. (b)

Let $X$ be a nonsingular projective toric compactification of $(\mathbb{C}^{*})^{n}$. Let $\overline{Y}$ be the Zariski closure of $Y$ in $X$, and suppose that $\overline{Y}$ is in general position with the boundary of $(\mathbb{C}^{*})^{n}$ in $X$. Then

[TABLE]

Here, we say that $\overline{Y}\subset X$ is in general position with the boundary $X\setminus(\mathbb{C}^{*})^{n}$ if $\overline{Y}$ does not contain any point of intersection of $n$ distinct irreducible components of $X\setminus(\mathbb{C}^{*})^{n}$.

Alternatively, the counting function $N_{\bf g}(Y,r)$ can be expressed as a counting function of common zeros of functions obtained by composing ${\bf g}$ with polynomials generating the defining ideal of $Y$. In this context, we prove the following:

Theorem 1.3**.**

Let $F,\,G\in\mathbb{C}[x_{1},\dots,x_{n}]$ be nonconstant coprime polynomials. Let $g_{1},\ldots,g_{n}$ be entire functions without zeros. Assume that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin\mathbb{C}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$. Let $\epsilon>0$.

1. (a)

Then

[TABLE] 2. (b)

If $F$ and $G$ do not both vanish at the origin $(0,\ldots,0)$, then

[TABLE]

Here, as will be discussed in more detail later, $N_{\gcd}(f,g,r)$ and $T_{\gcd}(f,g,r)$ are analogues of the greatest common divisor of two integers. The function $N_{\gcd}(f,g,r)$ is simply the counting function of common zeros of $f$ and $g$.

A particularly simple consequence of Theorem 1.3 is the following result.

Corollary 1.4**.**

Let $f$ and $g$ be multiplicatively independent non-constant entire functions without zeros. Then for any $\epsilon>0$,

[TABLE]

An elementary proof of Corollary 1.4 was previously obtained in [14] by adapting the number-theoretical arguments of [4] to Nevanlinna theory.

It is natural to try to extend Theorem 1.3 (and Corollary 1.4) in an appropriate way to entire functions, or more generally, to meromorphic functions. In this direction, by adapting the ideas and methods of Silverman [19] to a pair of meromorphic functions, the following estimate was established by Pasten and the second author in [14, Proposition 7.2] under the assumption of Vojta’s conjecture for a blow-up of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at a single point: If $f$ and $g$ are algebraically independent complex meromorphic functions, then for all $\epsilon>0$,

[TABLE]

Again under the assumption of Vojta’s conjecture (as for (2)), the following asymptotic gcd estimate is formulated in [14, Proposition 7.4]: If $f$ and $g$ are multiplicatively independent meromorphic functions, then for any $\epsilon>0$, there exists $n_{0}$ such that for all $n\geq n_{0}$,

[TABLE]

A second goal of this article is to prove (unconditional) asymptotic gcd estimates in a more general context:

Theorem 1.5**.**

Let $F,\,G\in\mathbb{C}[x_{1},\dots,x_{n}]$ be nonconstant coprime polynomials such that not both of them vanish at $(0,\dots,0)$. Let $g_{1},\ldots,g_{n}$ be meromorphic functions such that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin\mathbb{C}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$. Then for any $\epsilon>0$, there exists $k_{0}$ such that for all $k\geq k_{0}$,

1. (a)

[TABLE] 2. (b)

[TABLE]

if $g_{1},\ldots,g_{n}$ are entire functions.

In particular, we prove the conjectured inequality (4):

Corollary 1.6**.**

Let $f$ and $g$ be multiplicatively independent meromorphic functions. Then for any $\epsilon>0$, there exists $k_{0}$ such that for all $k\geq k_{0}$,

[TABLE]

When $f$ and $g$ are algebraically independent meromorphic functions, Corollary 1.6 was recently obtained by Guo and the second author in [8] with $\epsilon$ replaced by $\frac{1}{2}+\epsilon$. We refer to [14] for further discussion of gcd problems for both complex and non-archimedean meromorphic functions.

Even in the special case when $g_{1},\ldots,g_{n}$ are complex polynomials, Theorem 1.5 gives new results.

Remark 1.7**.**

Let $F,\,G\in\mathbb{C}[x_{1},\dots,x_{n}]$ be nonconstant coprime polynomials such that not both of them vanish at $(0,\dots,0)$. Let $g_{1},\ldots,g_{n}\in\mathbb{C}[z]$ be complex polynomials such that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin\mathbb{C}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$. It is elementary that in this case,

[TABLE]

and

[TABLE]

where the $\gcd$ on the right-hand side of (5) is the greatest common divisor in the polynomial ring $\mathbb{C}[z]$. Then Theorem 1.5 implies that for any $\epsilon>0$, there exists $k_{0}$ such that for all $k\geq k_{0}$,

[TABLE]

More generally, Theorem 1.5 gives a similar statement for rational functions $g_{1},\ldots,g_{n}\in\mathbb{C}(z)$. When $n>2$, the only previous result in this direction appears to be a result of Ostafe [13, Th. 1.3], which considers special polynomials such as $F=x_{1}\cdots x_{r}-1,G=x_{r+1}\cdots x_{n}-1$, but proves a stronger uniform bound independent of $k$. It is noted in [13] that it appears to be difficult to extend the techniques used there to obtain results for general $F$ and $G$. In the $n=2$ case, previous results include the original theorem of Ailon-Rudnick [1] in this setting and extensions of Ostafe [13] (both with uniform bounds).

We will use Theorem 1.3 and Theorem 1.5 to solve the following quotient problem, which can be considered an analogue of the“Hadamard quotient theorem” for recurrence sequences proved by van der Poorten (see [17] and [20], and see [3] and [25] for an overview of the existing improvements). To state the result we make the following definitions: Let $G\in\mathbb{C}[x_{1},\dots,x_{n}]$ be a nonconstant polynomial such that $G(0,\dots,0)\neq 0$. Since $G$ has a non-zero constant term, after arranging the index set in some order, we may write

[TABLE]

where $a_{{\bf i}(j)}\neq 0$ for $0\leq j\leq\ell$. Then for entire functions $g_{1},\ldots,g_{n}$, we let

[TABLE]

For functions $h_{i}:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$, $i=1,2$, we write $h_{1}(r)\asymp h_{2}(r)$ if there exist positive numbers $a$ and $b$ such that $ah_{1}(r)\leq h_{2}(r)\leq bh_{1}(r)$ for all sufficiently large $r$.

Corollary 1.8**.**

Let $F,\,G\in\mathbb{C}[x_{1},\dots,x_{n}]$ be nonconstant coprime polynomials such that $G(0,\dots,0)\neq 0$. Let $g_{1},\ldots,g_{n}$ be entire functions such that $T_{\mathfrak{g}_{G}}(r)\asymp\max_{1\leq i\leq n}\{T_{g_{i}}(r)\}$.

1. (a)

If $F(g_{1}^{k},\ldots,g_{n}^{k})/G(g_{1}^{k},\ldots,g_{n}^{k})$ is an entire functions for infinitely many positive integers $k$; or 2. (b)

$g_{1},\ldots,g_{n}$* are entire functions without zeros and $F(g_{1},\ldots,g_{n})/G(g_{1},\ldots,g_{n})$ is an entire function,*

then there exists an index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$ such that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\in\mathbb{C}$.

Remark 1.9**.**

The case where $(g_{1}^{k}-1)/(g_{2}^{k}-1)$ is entire for infinitely many positive integers $k$ (i.e., $F=-1+x_{1}$ and $G=-1+x_{2}$) is treated in [8] under the assumption that $T_{g_{1}}(r)\asymp T_{g_{2}}(r)$. The case $G=1+a_{1}x_{1}+\ldots+a_{n}x_{n}$, $F=b_{0}+b_{1}x_{n+1}+\ldots+b_{m}x_{n+m}$ ($a_{i}b_{j}\neq 0$ for $1\leq i\leq n$ and $0\leq j\leq m$) with the assumption that $\max\{T_{g_{1}}(r),\ldots,T_{g_{n}}(r)\}\asymp\max\{T_{g_{n+1}}(r),\ldots,T_{g_{n+m}}(r)\}$ is studied in [6] when $a_{i}$ and $b_{j}$ are constants and in [7] when $a_{i}$ and $b_{j}$ are “small functions”, with a proof obtained by adapting the argument of [3], where Corvaja and Zannier proved a stronger version of the Hadamard quotient theorem through a sophisticated application of Schmidt’s subspace theorem. The result in [6] can be obtained from Corollary 1.8 since in this situation

[TABLE]

and

[TABLE]

Remark 1.10**.**

The growth condition on characteristic functions is essential, at least in part (b). For instance, let $g(z)=\exp(2\pi iz)$ and $f(z)=\exp(2\pi ip(z))$, where $p(x)\in\mathbb{Z}[x]$ is a polynomial of degree at least $2$. Then $(g-1)|(f-1)$, but $f$ and $g$ are algebraically independent.

Finally, we will consider the case when the coefficients of $F$ and $G$ are functions. More precisely, let ${\bf g}$ be a holomorphic map from $\mathbb{C}$ to $\mathbb{P}^{n}$. We say a meromorphic function $a$ is a small function with respect to ${\bf g}$ if $T_{a}(r)={\rm o}(T_{\bf g}(r))$. Let $K_{\bf g}$ be the field containing all small functions with respect to ${\bf g}$. Let $F,G\in K_{\bf g}[x_{1},\ldots,x_{n}]$ be nonconstant coprime polynomials. We establish results analogous to Theorem 1.3, Theorem 1.5, and Corollary 1.8 in this situation. The results are stated in Section 5.

The proofs of Theorem 1.2 and Theorem 1.3 are inspired by recent work of the first author [10] on analogous inequalities in Diophantine approximation involving greatest common divisors of multivariable polynomials evaluated at $S$-unit arguments. The proofs of Theorem 1.5 and Corollary 1.6 go well beyond what is possible in the arithmetic setting, as they take advantage of inequalities involving truncated counting functions and Wronskian terms, whose analogues are still largely conjectural in Diophantine approximation.

It will be useful to give a brief overview of the analogous results in Diophantine approximation involving greatest common divisors. The first general result in this direction is due to Bugeaud, Corvaja, and Zannier, who in 2003 proved:

Theorem 1.11** (Bugeaud, Corvaja, Zannier [2]).**

Let $a,b\in\mathbb{Z}$ be multiplicatively independent integers. Then for every $\epsilon>0$,

[TABLE]

for all but finitely many positive integers $n$.

Let $k$ be a number field, $M_{k}$ the set of places of $k$, $S\subset M_{k}$ a finite set of places containing the archimedean places, $\mathcal{O}_{k,S}$ the ring of $S$-integers of $k$, and $\mathcal{O}_{k,S}^{*}$ the group of $S$-units. Let $\mathbb{G}_{m}^{n}$ be the $n$-dimensional algebraic torus and $\mathbb{G}_{m}^{n}(\mathcal{O}_{k,S})=(\mathcal{O}_{k,S}^{*})^{n}$. To state more general results, for $\alpha,\beta\in k$, we define the generalized logarithmic greatest common divisor and the $\gcd$ counting function (depending on a choice of $S$), respectively, by

[TABLE]

where $|\cdot|_{v}$ is an appropriately normalized absolute value associated to $v$, $h$ is the standard (absolute logarithmic) Weil height on projective space, and $\log^{-}x=\min\{0,\log x\}$ (see [10] for details). After work of Corvaja and Zannier [4] in the $2$-dimensional case, the first author generalized Theorem 1.11 as follows.

Theorem 1.12** (Corvaja, Zannier [4] ($n=2$), Levin [10] ($n>2$)).**

Let $F,G\in k[x_{1},\ldots,x_{n}]$ be coprime polynomials. Let $\epsilon>0$.

1. (a)

There exists a finite union $Z$ of translates of proper algebraic subgroups of $\mathbb{G}_{m}^{n}$ such that

[TABLE]

for all $(u_{1},\ldots,u_{n})\in\mathbb{G}_{m}^{n}(\mathcal{O}_{k,S})\setminus Z$. 2. (b)

Suppose additionally that not both of $F$ and $G$ vanish at the origin $(0,0,\ldots,0)$. Then there exists a finite union $Z$ of translates of proper algebraic subgroups of $\mathbb{G}_{m}^{n}$ such that

[TABLE]

for all $(u_{1},\ldots,u_{n})\in\mathbb{G}_{m}^{n}(\mathcal{O}_{k,S})\setminus Z$.

One can also give a geometric version of Theorem 1.12.

Theorem 1.13**.**

Let $\mathbb{G}_{m}^{n}\subset X$ be a nonsingular projective toric variety of dimension $n$, and let $Y$ be a closed subscheme of $X$ of codimension at least $2$, both defined over a number field $k$. Let $A$ be a big divisor on $X$. Let $S$ be a finite set of places of $k$ containing the archimedean places. Let $\epsilon>0$.

1. (a)

There exists a finite union $Z$ of translates of proper algebraic subgroups of $\mathbb{G}_{m}^{n}$ such that

[TABLE]

for all $P\in\mathbb{G}_{m}^{n}(\mathcal{O}_{k,S})\setminus Z\subset X(k)$. 2. (b)

Suppose that $Y$ is in general position with the boundary of $\mathbb{G}_{m}^{n}$ in $X$. Then there exists a finite union $Z$ of translates of proper algebraic subgroups of $\mathbb{G}_{m}^{n}$ such that

[TABLE]

for all $P\in\mathbb{G}_{m}^{n}(\mathcal{O}_{k,S})\setminus Z\subset X(k)$.

All of the described arithmetic inequalities rely on Schmidt’s Subspace Theorem in Diophantine approximation. The proofs of Theorem 1.2 and Theorem 1.3 are adapted from the proofs in [10], and use the analogue of Schmidt’s Subspace Theorem in Nevanlinna theory: Cartan’s second main theorem (in a form due to Vojta). The proofs of the asymptotic gcd inequalities of Theorem 1.5 and Corollary 1.6 combine techniques from [10] along with new ideas in order to take advantage of the stronger error terms known in the second main theorem in Nevanlinna theory.

After giving the relevant background material in the next section, in Section 3 we prove the key technical results underlying the paper. In Section 4, we apply these results to prove our main theorems. In Section 5, we review some fundamental results in Nevanlinna theory with moving targets, and we state and prove the main theorems when the coefficients of $F$ and $G$ are small functions.

2. Background material

2.1. Nevanlinna theory over $\mathbb{C}$

We will set up some notation and definitions in Nevanlinna theory for complex meromorphic functions and recall some basic results. We refer to [9, Chapter VI ] or [16, Chapter 1] for details.

Let $f$ be a meromorphic function and $z\in\mathbb{C}$. Define $v_{z}(f):={\rm ord}_{z}(f)$,

[TABLE]

Let $n_{f}(\infty,r)$ (respectively, ${n}^{(Q)}_{f}(\infty,r)$) denote the number of poles of $f$ in $\{z:|z|\leq r\}$, counting multiplicity (respectively, ignoring multiplicity larger than $Q\in\mathbb{N}$). The counting function and truncated counting function of $f$ of order $Q$ at $\infty$ are defined respectively by

[TABLE]

and

[TABLE]

Then define the counting function $N_{f}(r,a)$ and the truncated counting function $N^{(Q)}_{f}(r,a)$ for $a\in\mathbb{C}$ as

[TABLE]

The proximity function $m_{f}(\infty,r)$ is defined by

[TABLE]

where $\log^{+}x=\max\{0,\log x\}$ for $x\geq 0$. For any $a\in\mathbb{C},$ the proximity function $m_{f}(a,r)$ is defined by

[TABLE]

Finally, the characteristic function is defined by

[TABLE]

We recall the following version of Jensen’s formula.

Lemma 2.1**.**

Let $f$ be a meromorphic function on $\{z:|z|\leq r\}$ which is not the zero function. Then

[TABLE]

where $c_{f}$ is the leading coefficient of $f$ expanded as Laurent series in $z$, i.e., $f=c_{f}z^{m}+\cdots$ with $c_{f}\neq 0$.

Jensen’s formula implies the first main theorem of Nevanlinna theory.

Theorem 2.2** (First Main Theorem).**

Let $f$ be a non-constant meromorphic function on $\mathbb{C}$. Then for every $a\in\mathbb{C}$, and any positive real number $r$,

[TABLE]

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

2.2. Nevanlinna theory for Cartier divisors

We recall some notation and properties from [22, Section 9 and 12]. Let $D$ be a Cartier divisor on a complex variety $X$. A Weil function for $D$ is a function $\lambda_{D}:(X\setminus\operatorname{Supp}D)(\mathbb{C})\to\mathbb{R}$ such that for all $x\in X(\mathbb{C})$, there is an open neighborhood $U$ of $x$ in $X$, a nonzero function $f\in K(X)$ such that $D|_{U}=(f)$, and a continuous function $\alpha:U(\mathbb{C})\to\mathbb{R}$ such that

[TABLE]

for all $x\in(U\setminus\operatorname{Supp})(\mathbb{C})$. We note that when $D$ is effective $\lambda_{D}$ can be extended to a function $X\to\mathbb{R}\cup\{\infty\}$. Let $f:\mathbb{C}\to X$ be a holomorphic map whose image is not contained in the support of the divisor $D$ on $X$. The proximity function of $f$ with respect to $D$ is defined by

[TABLE]

Let $n_{f}(D,t)$ (respectively, $n^{(Q)}_{f}(D,t)$) be the number of zeros of $\rho\circ f$ inside $\{|z|<t\}$, counting multiplicity, (respectively, ignoring multiplicity larger than $Q\in\mathbb{N}$) with $\rho$ a local defining function for $D$. The counting function and truncated counting function of $f$ of order $Q$ at $\infty$ are defined, respectively, by

[TABLE]

and

[TABLE]

The characteristic function relative to $D$ is defined, up to $O(1)$, by

[TABLE]

The following is the first main theorem.

Theorem 2.3**.**

Let $D$ and $D^{\prime}$ be Cartier divisors on $X$ whose supports do not contain the image of $f$, and suppose that $D^{\prime}$ is linearly equivalent to $D$. Then

[TABLE]

In particular, let $D$ be a hypersurface in $\mathbb{P}^{n}(\mathbb{C})$ defined by a homogeneous polynomial $F$ of degree $d$. The Weil function for $D$ can be taken as

[TABLE]

for ${\bf x}=[x_{0}:\cdots:x_{n}]\in\mathbb{P}^{n}(\mathbb{C})$. Let ${\mathbf{f}}:\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic map and $(f_{0},\dots,f_{n})$ be a reduced representation of ${\mathbf{f}}$, i.e. $f_{0},\dots,f_{n}$ are entire functions on $\mathbb{C}$ without common zeros such that for all $z\in\mathbb{C}$ we have ${\mathbf{f}}(z)=[f_{0}(z):\cdots:f_{n}(z)]$. The characteristic function $T_{\mathbf{f}}(r)$ is defined by

[TABLE]

where $\|{\mathbf{f}}(z)\|=\max\{|f_{0}(z)|,\dots,|f_{n}(z)|\}$. This definition is independent, up to an additive constant, of the choice of the reduced representation of ${\mathbf{f}}$. In this context, the First Main Theorem reads as follows.

Theorem 2.4**.**

Let ${\mathbf{f}}:\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic map, and let $D$ be a hypersurface in $\mathbb{P}^{n}(\mathbb{C})$ of degree $d$. If $f(\mathbb{C})\not\subset D$, then for $r>0$,

[TABLE]

where $O(1)$ is bounded independently of $r$.

We will make use of the following elementary inequality.

Lemma 2.5**.**

Let $g_{1},\ldots,g_{n}$ be meromorphic functions. Let ${\bf g}:=[1:g_{1}:\cdots:g_{n}]:\mathbb{C}\to\mathbb{P}^{n}.$ Then

[TABLE]

Proof.

Let $H$ be the hyperplane in $\mathbb{P}^{n}$ defined by $x_{0}=0$. Then clearly

[TABLE]

and

[TABLE]

After integrating, it follows from the definitions that

[TABLE]

∎

We now recall the following general form of the second main theorem from [16, Theorem A3.1.3] which was proved by Vojta in [21, Theorem 1] and Ru in [15, Theorem 2.3].

Theorem 2.6**.**

Let $f=(f_{0},\ldots,f_{n}):\mathbb{C}\to\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained in any proper linear subspace, and where $f_{0},\dots,f_{n}$ are entire functions with no common zeros. Let $H_{1},\dots,H_{q}$ be arbitrary hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$. Denote by $W(f)$ the Wronskian of $f_{0},\dots,f_{n}$. Then for any $\varepsilon>0$, we have the inequality

[TABLE]

where the maximum is taken over all subsets $J$ of $\{1,\dots,q\}$ such that the hyperplanes $H_{j}$, $j\in J$, are in general position.

Finally, we recall Cartan’s second main theorem with truncated counting functions. (See [16, Theorem A3.2.2].)

Theorem 2.7**.**

Let $H_{1},\dots,H_{q}$ be hyperplanes in $\mathbb{P}^{n}(\mathbb{C})$ in general position. Let $f=(f_{0},\ldots,f_{n}):\mathbb{C}\to\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve whose image is not contained in any proper linear subspace. Then for any $\varepsilon>0$, we have the following inequality:

[TABLE]

2.3. Counting functions and proximity functions for closed subschemes

We first recall some basic properties of Weil functions associated to closed subschemes from [18, Section 2]. Let $Y$ be a closed subscheme on a projective variety $X$ defined over $\mathbb{C}$. Then one can associate to it a function

[TABLE]

satisfying some functorial properties (up to a O(1)) analogous to the arithmetic case as described in [18, Theorem 2.1]. Intuitively, for each point $P\in X(\mathbb{C})$,

[TABLE]

The following lemma will be used to define Weil functions for closed subschemes.

Lemma 2.8**.**

Let $Y$ be a closed subscheme of $X$. There exist effective divisors $D_{1},\cdots,D_{r}$ such that

[TABLE]

Proof.

See Lemma 2.2 from [18]. ∎

Definition 2.9**.**

Let $X$ be a projective variety over $\mathbb{C}$ and let $Y\subset X$ be a closed subscheme of $X$. We define the Weil function for $Y$ as

[TABLE]

where $Y=\cap D_{i}$ (such $D_{i}$ exist according to the above lemma).

As usual, the Weil function $\lambda_{Y}$ is well-defined up to $O(1)$. Let $f:\mathbb{C}\to X$ be an analytic map. The proximity function of $f$ with respect to $Y$ is defined by

[TABLE]

the counting function of $f$ with respect to $Y$ is defined by

[TABLE]

where $n_{f}(Y,t)$ is the minimum of the number of zeros of $\rho_{i}\circ f$, $1\leq i\leq m$ inside $\{|z|<t\}$, counting multiplicity, with $\rho_{i}$, $1\leq i\leq m$, being local defining functions of $Y$. We note that the definition of the counting function extends without difficulty to the case when $Y$ is a closed subscheme of a quasi-projective variety. The characteristic function of $f$ with respect to $Y$ is defined by

[TABLE]

The proximity functions and counting functions of closed subschemes of $X$ satisfy additivity and functoriality properties as in the classical setting of Cartier divisors.

Recall the following definition which gives an analogue of the notion of gcd in the context of meromorphic functions. Let $f$ and $g$ be meromorphic functions. We write

[TABLE]

and

[TABLE]

Let $F,G\in\mathbb{C}[x_{0},\cdots,x_{n}]$ be coprime homogeneous polynomials and let $Y$ be the closed subscheme of $\mathbb{P}^{n}$ defined by the ideal $I=(F,G)$. Let ${\mathbf{f}}=(f_{0},\dots,f_{n}):\mathbb{C}\rightarrow\mathbb{P}^{n}(\mathbb{C})$, where $f_{0},\ldots,f_{n}$ are entire functions with no common zeros. The counting function of $Y$ agrees with the gcd counting function of $F({\mathbf{f}})$ and $G({\mathbf{f}})$, i.e.,

[TABLE]

We also define gcd proximity and characteristic functions, in analogy with (6) and (7). Let

[TABLE]

We have the expected relationship between $m_{\gcd},N_{\gcd},$ and $T_{\gcd}$.

Lemma 2.10**.**

Let $f$ and $g$ be meromorphic functions. Then

[TABLE]

Proof.

Let $h_{0}$ be an entire function such that $(h_{0},fh_{0},gh_{0})$ is a reduced representation of $[1:f:g]$, i.e., $h_{0},fh_{0},$ and $gh_{0}$ are entire and have no common zeros. Similarly, let $h_{1}$ be an entire function such that $[fh_{0}/h_{1},gh_{0}/h_{1}]$ is a reduced representation of $[f:g]$. Then from the definitions,

[TABLE]

It is elementary that for any real numbers $a,b>0$,

[TABLE]

Then we find that

[TABLE]

where the second line follows from the definition of $m_{\gcd}$ and Jensen’s formula. To complete the proof, we note that from its definition, one easily finds that $N_{h_{1}}(0,r)=N_{\gcd}(f,g,r)$. ∎

2.4. Polynomial rings and monomial orderings

Let $A=\mathbb{C}[x_{0},\ldots,x_{n}]$ be the polynomial ring in $n+1$ variables over $\mathbb{C}$. For ${\bf i}=(i_{0},\ldots,i_{n})\in\mathbb{N}^{n+1}$, we define

[TABLE]

and write

[TABLE]

Let

[TABLE]

be the set of monomials of $A$. We note that we use the convention that ${\mathbb{N}}$ is the set of nonnegative integers.

Recall that a monomial ordering on $A$ is a total ordering $>$ on Mon such that

1. (a)

If ${\bf x}^{\bf i}>{\bf x}^{\bf j}$, then ${\bf x}^{\bf i}{\bf x}^{\bf k}>{\bf x}^{\bf j}{\bf x}^{\bf k}$ for all ${\bf i}$, ${\bf j}$, ${\bf k}\in{\mathbb{N}^{n+1}}$. 2. (b)

${\bf x}^{\bf i}\geq 1$ for all ${\bf i}\in{\mathbb{N}^{n+1}}$.

We describe two monomial orderings that we will use. The lexicographic ordering on $A$ is the monomial ordering $>_{\rm lex}$ such that ${\bf x}^{\bf i}>_{\rm lex}{\bf x}^{\bf j}$ if the left-most non-zero entry of ${\bf i}-{\bf j}$ is positive. Let ${\bf u}\in{\mathbb{N}^{n+1}}$. We call ${\bf u}$ a weight vector and define the weight order associated to $\bf u$ as follows:

[TABLE]

If $>$ is a monomial ordering and $F\in\mathbb{C}[x_{0},\ldots,x_{n}]$ is a nonzero polynomial, we let ${\rm TM}(F)$ denote the trailing monomial of $F$ (the smallest monomial appearing in $F$ with a nonzero coefficient). If $F$ and $G$ are nonzero polynomials, then ${\rm TM}(FG)={\rm TM}(F){\rm TM}(G)$. For every nonnegative integer $m$ and subset $T\subset\mathbb{C}[x_{0},\ldots,x_{n}]$, we let

[TABLE]

and let ${\rm TM}(T)=\{{\rm TM}(F):F\in T\}$. We will use the following key lemma.

Lemma 2.11**.**

Let $F_{1},F_{2}\in k[x_{0},\ldots,x_{n}]$ be coprime homogeneous polynomials of degree $d$, where $k$ is a field. Let $I_{1}$, $I_{2}$, $I_{3}$ be the principal ideals $I_{1}=(F_{1})$, $I_{2}=(F_{2})$, $I_{3}=(F_{1}F_{2})$. Let $m\geq d$ and let $V=(F_{1},F_{2})_{m}:=k[x_{0},\ldots,x_{n}]_{m}\cap(F_{1},F_{2})$, a $k$-subspace of the ideal $(F_{1},F_{2})$. Let $V_{j}=(F_{1},F_{2})_{m}\cap I_{j}$, $j=1,2,3$. Let

[TABLE]

Then

[TABLE]

is a basis for $(F_{1},F_{2})_{m}$. Moreover,

[TABLE]

for $j=1,2$, and

[TABLE]

Proof.

Since $f_{1}$ and $f_{2}$ are coprime, we have $I_{1}\cap I_{2}=I_{3}$ and $V_{1}\cap V_{2}=V_{3}$. For any finite-dimensional subspace $W\subset k[x_{0},\ldots,x_{n}]$, it is easy to see that $\#{\rm TM}(W)=\dim W$. Since $V=V_{1}+V_{2}$ and $\dim V=\dim V_{1}+\dim V_{2}-\dim V_{3}$, we have $\dim V=\#B_{1}+\#B_{2}-\#B_{1}^{\prime}$. Let $V_{1}^{\prime}$ be the span of $B_{1}\setminus B_{1}^{\prime}$. Then ${\rm TM}(V_{1}^{\prime})={\rm TM}(B_{1}\setminus B_{1}^{\prime})={\rm TM}(B_{1})\setminus{\rm TM}(B_{1}^{\prime})$. If $p\in V_{1}^{\prime}\cap V_{2}$, $p\neq 0$, then $p\in V_{3}$ and ${\rm TM}(p)\in{\rm TM}(V_{3})={\rm TM}(B_{1}^{\prime})$, a contradiction. So $V_{1}^{\prime}\cap V_{2}=0$. Since $\dim V_{1}^{\prime}+\dim V_{2}=\#B_{1}-\#B_{1}^{\prime}+\#B_{2}=\dim V$ as well, it follows that $B=(B_{1}\setminus B_{1}^{\prime})\cup B_{2}$ is a basis for $V$.

As is well-known, the number of monomials of degree $\delta$ in $x_{0},\ldots,x_{n}$ is $\binom{n+\delta}{n}$. Then

[TABLE]

By symmetry, we have

[TABLE]

for $i=0,\ldots,n$. This yields the first summation formula. The second summation formula follows similarly. ∎

3. Key Theorem

In this section, we prove a fundamental result underlying the proofs of our main theorems.

Theorem 3.1**.**

Let $F,G\in\mathbb{C}[x_{0},\cdots,x_{n}]$ be coprime homogeneous polynomials of the same degree $d>0$. Let $I$ be the set of exponents ${\bf i}$ such that ${\bf x}^{\bf i}$ appears with a nonzero coefficient in either $F$ or $G$. Let $m\geq d$ be a positive integer. Let $g_{0},g_{1},\dots,g_{n}$ be entire functions without common zeros such that the set $\{g_{0}^{i_{0}}\cdots g_{n}^{i_{n}}:i_{0}+\cdots+i_{n}=m\}$ is linearly independent over $\mathbb{C}$. Then for any $\epsilon>0$, there exists a positive integer $L$ such that the following holds:

[TABLE]

where ${\bf g}=(g_{0},g_{1},\dots,g_{n}):\mathbb{C}\to\mathbb{P}^{n}$, $c_{m,n,d}=2\binom{m+n-d}{n+1}-\binom{m+n-2d}{n+1}$, $M=M_{m,n,d}=2\binom{m+n-d}{n}-\binom{m+n-2d}{n}$, and $M^{\prime}$ is an integer of order $O(m^{n-2})$.

Proof of Theorem 3.1.

Let $\{\phi_{1},\ldots,\phi_{M}\}$ be a basis of the $\mathbb{C}$-vector space $(F,G)_{m}$, where $M=\dim(F,G)_{m}$. For each $z\in\mathbb{C}$, we construct a basis $B_{z}$ for $V_{m}=\mathbb{C}[x_{0},\ldots,x_{n}]_{m}/(F,G)_{m}$ as follows. For ${\bf i}=(i_{0},\cdots,i_{n})\in\mathbb{N}^{n+1}$, we let ${\bf g}(z)^{\bf i}=g_{0}(z)^{i_{0}}\cdots g_{n}(z)^{i_{n}}$. Choose a monomial ${\bf x}^{{\bf i}_{1}}\in\mathbb{C}[x_{0},\ldots,x_{n}]_{m}$ so that $|{\bf g}(z)^{{\bf i}_{1}}|$ is minimal subject to the condition ${\bf x}^{{\bf i}_{1}}\notin(F,G).$ Suppose now that ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{j}}$ have been constructed and are linearly independent modulo $(F,G)_{m}$, but don’t span $\mathbb{C}[x_{0},\ldots,x_{n}]_{m}$ modulo $(F,G)_{m}$. Then we let ${\bf x}^{{\bf i}_{j+1}}\in\mathbb{C}[x_{0},\ldots,x_{n}]_{m}$ be a monomial such that $|{\bf g}(z)^{{\bf i}_{j+1}}|$ is minimal subject to the condition that ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{j+1}}$ are linearly independent modulo $(F,G)_{m}$. In this way, we construct a basis of $V_{m}$ with monomial representatives ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{M^{\prime}}}$, where $M^{\prime}=\dim V_{m}$. Let $I_{z}=\{{\bf i}_{1},\ldots,{\bf i}_{M^{\prime}}\}$. Then for each ${\bf i}$, $|{\bf i}|=m$, we have

[TABLE]

for some choice of coefficients $c_{{\bf i},j}\in\mathbb{C}$. Then for each such ${\bf i}$ there is a linear forms $L_{z,{\bf i}}$ over $\mathbb{C}$ such that

[TABLE]

We note that since there are only finitely many choices of a monomial basis of $V_{m}$, there are only finitely many choices of $c_{{\bf i},j}$, even as $z$ runs through all of $\mathbb{C}$. Note also that $\{L_{z,{\bf i}}(\phi_{1},\ldots,\phi_{M})\,|\,|{\bf i}|=m,{\bf i}\notin I_{z}\}$ is a basis for $(F,G)_{m}$. From the definition of ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{M^{\prime}}}$, we have the key inequality

[TABLE]

where $\Phi({\bf g}(z))=(\phi_{1}({\bf g}(z)),\ldots,\phi_{M}({\bf g}(z)))$, and the constant $C$ is independent of $z$ as the choice of $c_{{\bf i},j}$ is finite. The map $(\phi_{1}({\bf g}),\ldots,\phi_{M}({\bf g}))$ may not be a reduced presentation of $\Phi({\bf g})$. Let $h$ be an entire function such that $F({\bf g})/h$ and $G({\bf g})/h$ are entire and have no common zeros, i.e., $h$ is a gcd of $F({\bf g})$ and $G({\bf g})$. Let $\psi_{i}:=\phi_{i}({\bf g})/h$, $1\leq i\leq M$. As $\phi_{i}\in(F,G)$, the function $\psi_{i}$ is entire for $1\leq i\leq M$. Moreover, since $FX_{i}^{m-d},GX_{i}^{m-d}\in(F,G)_{m}$, $0\leq i\leq n$, and $g_{0},\ldots,g_{n}$ have no common zero, the functions $\psi_{i}$, $1\leq i\leq M$, have no common zero. Hence $\Psi:=(\psi_{1},\ldots,\psi_{M})$ is a reduced form of $\Phi({\bf g})$.

Applying Theorem 2.6, the second main theorem, to the map $\Psi({\bf g}(z))$ with the choice of linear forms $L_{z,{\bf i}}$, $|{\bf i}|=m$, ${\bf i}\notin I_{z}$, $z=re^{i\theta}\in\mathbb{C}$, we get for any $\epsilon>0$,

[TABLE]

Next, we will derive a lower bound for the left hand side of (9). By definition of the characteristic function, Lemma 2.1 and the choice of $h$, we have

[TABLE]

On the other hand, since $\phi_{i}\in\mathbb{C}[x_{0},\ldots,x_{n}]_{m}$,

[TABLE]

and hence

[TABLE]

In conclusion, we have

[TABLE]

To estimate the first term of (9), we use the key inequality (8) to derive

[TABLE]

where $c$ is a constant independent of $z$. Note that

[TABLE]

and

[TABLE]

By Lemma 2.1, Jensen’s formula, the integration of (12) from 0 to $2\pi$ over $d\theta$ gives

[TABLE]

It then follows from (9), (11), and (15) that

[TABLE]

By direction calculation, we find that $M^{\prime}=\binom{m+n}{n}-M=O(m^{n-2})$. Alternatively, since $F$ and $G$ are coprime, the ideal $(F,G)$ defines a closed subset of $\mathbb{P}^{n}$ of codimension at least 2, and it follows from the theory of Hilbert functions and Hilbert polynomials that $M^{\prime}=O(m^{n-2})$.

It’s clear from (17) that it suffices to show that there exists a large integer $L$ (to be determined later) such that

[TABLE]

The above inequality can be deduced from the inequality

[TABLE]

for all $z\in\mathbb{C}$. The inequality holds trivially if $v_{z}^{+}(g_{i})\leq L$ for each $0\leq i\leq n$. Therefore, we only need to consider the case where $v_{z}^{+}(g_{i})>L$ for some $0\leq i\leq n$.

For $z\in\mathbb{C}$, we define a monomial ordering $>_{{\bf g}(z)}$ on $A=\mathbb{C}[x_{0},\cdots,x_{n}]$ using the weight vector ${\bf u}=(v_{z}(g_{0}),\ldots,v_{z}(g_{n}))$. Let

[TABLE]

where $\{F_{1},F_{2}\}=\{F,G\}$ and ${\rm TM}_{{\bf g}(z)}(F_{2})\leq{\rm TM}_{{\bf g}(z)}(F_{1})$. By Lemma 2.11

[TABLE]

is a basis for $(F,G)_{m}$. Write $B=\{\beta_{1},\ldots,\beta_{M}\}$. Let $\eta_{j}=\beta_{j}({\bf g})/h$ (note that the $\beta_{j}$ depend on $z$.) From the definition of $>_{{\bf g}(z)}$ and $F_{2}$, it follows that

[TABLE]

where $I$ is the set of exponents ${\bf i}$ such that ${\bf x}^{\bf i}$ appears with a nonzero coefficient in either $F$ or $G$. Then by the second part of Lemma 2.11, we have for each $z\in\mathbb{C}$,

[TABLE]

where

[TABLE]

On the other hand, from the basic properties of Wronskians, we have

[TABLE]

Combining (21) and (23), we obtain that

[TABLE]

Let $L=\frac{1}{2}M(M-1)c_{m,n,d}^{-1}$. The assumption that $v_{z}^{+}(g_{i})>L$ for some $0\leq i\leq n$ implies that

[TABLE]

∎

4. Proof of the Main Theorems

We recall Borel’s lemma. (See [16, Theorem A.3.3.2]).

Lemma 4.1** (Borel’s Lemma).**

Let $f_{0},\ldots,f_{n+1}$ be entire functions without zeros, satisfying

[TABLE]

Define an equivalence relation $i\sim j$ if $f_{i}/f_{j}$ is constant. Then for each equivalence class $S$ we have

[TABLE]

We also recall the following result of Green [5] (See [9, Chapter VII, Theorem 4.1]).

Lemma 4.2**.**

Let $f_{0},\ldots,f_{n}$ be entire functions with no common zeros satisfying

[TABLE]

Suppose that none of the $f_{i}$ are identically [math]. Define an equivalence relation $i\sim j$ if $f_{i}/f_{j}$ is constant. If $k\geq n^{2}$, then for each equivalence class $S$ we have

[TABLE]

We will also make use of the following result on proximity functions.

Theorem 4.3**.**

Let $G\in\mathbb{C}[x_{1},\cdots,x_{n}]$ be a polynomial that does not vanish at the origin $(0,\ldots,0)$. Suppose that $g_{1},\dots,g_{n}$ are entire functions such that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin\mathbb{C}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$. For all $\epsilon>0$,

1. (a)

there exists a positive integer $k_{0}$ such that for all $k\geq k_{0}$,

[TABLE] 2. (b)

if in addition each $g_{i}$, $1\leq i\leq n$, has no zero, then

[TABLE]

Proof.

By Lemma 4.2, the set $\{{\bf g}^{k{\bf i}}:=g_{1}^{ki_{1}}\cdots g_{n}^{ki_{n}}\,|\,{\bf i}=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n},|{\bf i}|\leq d\}$ is linearly independent for $k\geq\binom{d+n}{n}^{2}$. Since $G(0,\ldots,0)\neq 0$, $G$ must have a non-zero constant term, and by arranging the index set in some order, we may write

[TABLE]

where $a_{{\bf i}(j)}\neq 0$ for $0\leq j\leq\ell$. Then we have

[TABLE]

We apply Theorem 2.7, Cartan’s truncated second main theorem, to the holomorphic map

[TABLE]

associated with the above expression ($k\geq\binom{d+n}{n}^{2}$), and with the set of hyperplanes given by the coordinate hyperplanes of $\mathbb{P}^{\ell}$ and the one defined by $\sum_{i=0}^{\ell}a_{{\bf i}(j)}X_{j}$. Then for any $\epsilon>0$, we have

[TABLE]

We note that

[TABLE]

where the last inequality is due to the definition of characteristic functions and that $\mathfrak{g}(k)=(1,{\bf g}^{k{\bf i}(1)},\ldots,{\bf g}^{k{\bf i}(\ell)})$. Then for $k>\max\{\frac{2\ell^{2}}{\epsilon},\binom{d+n}{n}^{2}\}$,

[TABLE]

Since $G(g_{1}^{k},\ldots,g_{n}^{k})$ is an entire function,

[TABLE]

Consequently,

[TABLE]

When the $g_{i}$ are entire functions without zeros, we may assume that the set $\{{\bf g}^{{\bf i}}:=g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\,|\,{\bf i}=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n},|{\bf i}|\leq d\}$ is linearly independent by Lemma 4.1. Then we can repeat the previous argument for $k=1$ with the additional condition $N_{g_{i}}(0,r)=O(1)$, $1\leq i\leq n$, to conclude the proof. ∎

Proof of Theorem 1.2 and Theorem 1.3.

We first claim that $g_{1},\dots,g_{n}$ are algebraically independent over $\mathbb{C}$ under the assumption that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin\mathbb{C}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$. If not, then they satisfy a non-trivial $\mathbb{C}$-linear relation, say

[TABLE]

where the sum is over finitely many index sets ${\bf i}=(i_{1},\dots,i_{n})$ and $a_{\bf i}\in\mathbb{C}^{*}$. We may further assume that no proper sub-sum of the left hand side of (26) is zero. Since $g_{1},\ldots,g_{n}$ are entire functions without zeros, Borel’s Lemma implies that we have a pair of indices ${\bf i}=(i_{1},\dots,i_{n})\neq{\bf j}=(j_{1},\dots,j_{n})$ such that $a_{\bf i}g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}$ is a constant multiple of some $a_{\bf j}g_{1}^{j_{1}}\cdots g_{n}^{j_{n}}$ appearing on the left hand side of (26), and hence $g_{1}^{i_{1}-j_{1}}\cdots g_{n}^{i_{n}-j_{n}}\in\mathbb{C}^{*}$, contradicting our assumption.

We prove part (a) of both theorems first. Consider $(\mathbb{C}^{*})^{n}\subset\mathbb{P}^{n}$, where we identify $(x_{1},\ldots,x_{n})\in(\mathbb{C}^{*})^{n}$ with $[1:x_{1}:\cdots:x_{n}]$, and let $\overline{Y}$ be the Zariski closure of $Y$ in $\mathbb{P}^{n}$. Then $N_{\bf g}(Y,r)=N_{\bf g}(\overline{Y},r)$, and we may assume that $Y$ is a closed subscheme of $\mathbb{P}^{n}$ (of codimension at least $2$).

Let $I\subset\mathbb{C}[x_{0},\ldots,x_{n}]$ be a homogeneous ideal associated to $Y$. We can find homogeneous polynomials $\tilde{F},\tilde{G}\in I$ of the same degree $d$ such that $\tilde{F}$ and $\tilde{G}$ are coprime. Let $Y^{\prime}$ be the closed subscheme defined by the ideal $(\tilde{F},\tilde{G})$. Then

[TABLE]

for all $r>0$. Therefore, we may assume that $Y$ is defined by the ideal $(\tilde{F},\tilde{G})$ after replacing $Y$ by $Y^{\prime}$. Then for any $\epsilon>0$, Theorem 3.1 for any (large) $m$ implies that

[TABLE]

as the $g_{i}$ are entire functions with no zeros. Let $\epsilon^{\prime}>0$. Since $M^{\prime}=O(m^{n-2})$ and $M=\frac{m^{n}}{n!}+O(m^{n-1})$, choosing $m$ large enough, depending only on $\epsilon^{\prime}$, (27) implies that

[TABLE]

This completes the proof of Theorem 1.2(a).

To show Theorem 1.3(a), we note that we may assume $\deg F=\deg G=d$ since

[TABLE]

where $e=\deg G$ and $h=\deg F$. Then Theorem 1.3(a) follows from (28) by taking

[TABLE]

and using Lemma 2.5.

Suppose in addition that, say, $G$ does not vanish at the origin $(0,\ldots,0)$. It is immediate from the definitions that

[TABLE]

and so Theorem 4.3 implies that

[TABLE]

Combined with part (a) above and Lemma 2.10, we obtain Theorem 1.3(b).

It remains to prove Theorem 1.2(b). Let $X$ be a nonsingular projective toric compactification of $(\mathbb{C}^{*})^{n}$. Let $\overline{Y}$ be the Zariski closure of $Y$ in $X$, and suppose that $\overline{Y}$ is in general position with the boundary of $(\mathbb{C}^{*})^{n}$ in $X$. Since $N_{\bf g}(Y,r)=N_{\bf g}(\overline{Y},r)$, by part (a) it suffices to show that $m_{\bf g}(\overline{Y},r)\leq_{\rm exc}\epsilon T_{\bf g}(r).$ In fact, we will show the stronger statement with $\overline{Y}$ replaced by an effective divisor $D$ on $X$ in general position with the boundary of $(\mathbb{C}^{*})^{n}$ in $X$. We follow the proof of [10, Theorem 4.4], which shows that there exist embeddings $\phi_{i}:(\mathbb{C}^{*})^{n}\to\mathbb{A}^{n}$ (which extend an automorphism of $(\mathbb{C}^{*})^{n}$) and polynomials $p_{i}\in\mathbb{C}[x_{1},\ldots,x_{n}]$ nonvanishing at the origin, $i=1,\ldots,t$, such that for every $P\in(\mathbb{C}^{*})^{n}\subset X(\mathbb{C})$, there exists $i\in\{1,\ldots,t\}$ satisfying

[TABLE]

Thus,

[TABLE]

for all $P\in(\mathbb{C}^{*})^{n}\subset X(\mathbb{C})$. By Theorem 4.3, this implies that for any $\epsilon>0$,

[TABLE]

completing the proof.

∎

Proof of Theorem 1.5.

By replacing $F$ and $G$ by suitable linear combinations of $F$ and $G$, we may assume that $F$ and $G$ have the same degree $d$, and that neither $F$ nor $G$ vanishes at the origin. We now consider the two related homogeneous polynomials

[TABLE]

Since $F$ and $G$ are coprime, it follows easily that $F_{1}$ and $G_{1}$ are coprime. Let $g_{1},\dots,g_{n}$ be meromorphic functions. Then there exists an entire function $h_{0}$ such that $h_{0}$ and $h_{i}:=g_{i}\cdot h_{0}$, $1\leq i\leq n$, are entire functions without a common zero. Therefore, $(h_{0},h_{1},\ldots,h_{n})$ is a reduced form of the holomorphic map ${\bf g}:=[1:g_{1}:\cdots:g_{n}]:\mathbb{C}\to\mathbb{P}^{n}$. Assume that $g_{1}^{j_{1}}\dots g_{n}^{j_{n}}$ is not constant for any $(j_{1},\dots,j_{n})\neq(0,\dots,0)\in\mathbb{Z}^{n}$. Let

[TABLE]

Then the set $\{(h_{0}^{k})^{i_{1}}\cdots(h_{n}^{k})^{i_{n}}:i_{0}+\cdots+i_{n}=m\}$ is linearly independent over $\mathbb{C}$ by Lemma 4.2. Let ${\bf h}^{k}=(h_{0}^{k},\cdots,h_{n}^{k}):\mathbb{C}\to\mathbb{P}^{n}$. Let $I$ be the set of exponents ${\bf i}$ such that ${\bf x}^{\bf i}$ appears with a nonzero coefficient in either $F_{1}$ or $G_{1}$. Note that

[TABLE]

and so for any $z\in\mathbb{C}$,

[TABLE]

since $x_{0}^{d}$ is a monomial appearing nontrivially in $F_{1}$ and $G_{1}$ as $F$ and $G$ don’t vanish at the origin. Then

[TABLE]

and it follows that

[TABLE]

Then by Theorem 3.1, for any $\epsilon>0$ there exists a positive integer $L$ such that for all positive integers $k$ satisfying (29), we have the following:

[TABLE]

where $c_{m,n,d}=2\binom{m+n-d}{n+1}-\binom{m+n-2d}{n+1}$, $M=2\binom{m+n-d}{n}-\binom{m+n-2d}{n}$, and $M^{\prime}$ is an integer of order $O(m^{n-2})$. Elementary computations give that

[TABLE]

Then

[TABLE]

Furthermore,

[TABLE]

and

[TABLE]

where $H_{i}$ is the coordinate hyperplane defined by $x_{i}=0$. Then we find that after choosing $m$ sufficiently large, for all sufficiently large $k$ (depending on $m$),

[TABLE]

This concludes the proof of Theorem 1.5 (a). If in addition $g_{1},\ldots,g_{n}$ are entire functions and, say, $G$ does not vanish at the origin, then

[TABLE]

and so Theorem 4.3 implies that

[TABLE]

for $k$ large enough. Together with (33), we reach the conclusion of (b). ∎

Proof of Corollary 1.6 .

If $f^{i}g^{j}\notin\mathbb{C}$ for any $(i,j)\neq(0,0)\in\mathbb{Z}^{2}$, then then the assertion follows from Theorem 1.5 for the meromorphic functions $f$ and $g$ with the polynomials $F=x_{1}-1$ and $G=x_{2}-1$. Suppose that $f^{i}g^{j}=c\in\mathbb{C}$ for some $(i,j)\in\mathbb{Z}^{2}\setminus\{(0,0)\}$. If $f^{k}-1$ and $g^{k}-1$ have no common zero for all $k$ sufficiently large, then the gcd inequality (4) holds trivially. Otherwise, we may find $z_{0}\in\mathbb{C}$ such that $f^{k}(z_{0})=g^{k}(z_{0})=1$ for some $k$. This implies that $c^{k}=1$ and hence $f^{ik}g^{jk}=1$, contradicting the assumption that $f$ and $g$ are multiplicatively independent. ∎

The proof of Corollary 1.4 is similar (with $k=1$) to the proof of Corollary 1.6, and so we omit it.

Proof of Corollary 1.8.

We prove part (a). The proof of part (b) is similar.

Suppose that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin\mathbb{C}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n+1}\setminus\{(0,\dots,0)\}$ and that $F(g_{1}^{k},\ldots,g_{n}^{k})/G(g_{1}^{k},\ldots,g_{n}^{k})$ is an entire function. Then

[TABLE]

and by Theorem 1.5 (a), for any $\epsilon>0$, there exists a positive integer $k_{0}$ such that

[TABLE]

if $k\geq k_{0}$. On the other hand, using the same notation as in the proof of Theorem 4.3 and recalling the first part of (25),

[TABLE]

for $k>k_{1}:=\max\{\frac{2\ell^{2}}{\epsilon},\binom{d+n}{n}^{2}\}$. Together with (34), we get

[TABLE]

for $k\geq\max\{k_{0},k_{1}\}$. Hence,

[TABLE]

contradicting the assumption that $T_{\mathfrak{g}_{G}}(r)\asymp\max_{1\leq i\leq n}\{T_{g_{i}}(r)\}$. ∎

5. GCD with Moving Targets

5.1. Statement of the main results

Let ${\bf g}$ be a holomorphic map from $\mathbb{C}$ to $\mathbb{P}^{n}$. Let $K_{\bf g}$ be the set containing all meromorphic functions $a$ such that $T_{a}(r)={\rm o}(T_{\bf g}(r))$. By the basic properties of characteristic functions, $K_{\bf g}$ is a field.

Theorem 5.1**.**

Let $g_{1},\ldots,g_{n}$ be entire functions without zeros and ${\bf g}=(1,g_{1},\dots,g_{n})$. Let $F,G\in K_{\bf g}[x_{1},\ldots,x_{n}]$ be nonconstant coprime polynomials. Assume that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin K_{\bf g}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$. Let $\epsilon>0$.

1. (a)

Then

[TABLE] 2. (b)

If $F$ and $G$ are not both identically zero at the origin $(0,\dots,0)$ and the coefficients of $F$ and $G$ are entire functions in $K_{\bf g}$, then

[TABLE]

Theorem 5.2**.**

Let $g_{1},\ldots,g_{n}$ be meromorphic functions and ${\bf g}=[1:g_{1}:\dots:g_{n}]$ a holomorphic map from $\mathbb{C}$ to $\mathbb{P}^{n}$. Let $F,G\in K_{\bf g}[x_{1},\ldots,x_{n}]$ be nonconstant coprime polynomials such that not both of them are identically zero at $(0,\dots,0)$. If $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin K_{\bf g}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$, then for any $\epsilon>0$, there exists $k_{0}$ such that for $k\geq k_{0}$

1. (a)

[TABLE] 2. (b)

[TABLE]

if $g_{1},\ldots,g_{n}$ are entire functions and the coefficients of $F$ and $G$ are entire functions in $K_{\bf g}$.

Let $g_{1},\ldots,g_{n}$ be entire functions and ${\bf g}=(1,g_{1},\dots,g_{n})$. Let $G\in K_{\bf g}[x_{1},\dots,x_{n}]$ be a nonconstant polynomial such that $G(0,\dots,0)$ is not identically zero. Since $G$ has a non-zero constant term, after arranging the index set in some order, we may write

[TABLE]

where $a_{{\bf i}(j)}\neq 0$ for $0\leq j\leq\ell$. We then let $\mathfrak{g}_{G}:=(1,{\bf g}^{{\bf i}(1)},\ldots,{\bf g}^{{\bf i}(\ell)}):\mathbb{C}\to\mathbb{P}^{\ell}.$

Corollary 5.3**.**

Let $g_{1},\ldots,g_{n}$ be entire functions and ${\bf g}=(1,g_{1},\dots,g_{n})$. Assume that $T_{\mathfrak{g}_{G}}(r)\asymp\max_{1\leq i\leq n}\{T_{g_{i}}(r)\}$. Let $F,\,G\in K_{\bf g}[x_{1},\dots,x_{n}]$ be nonconstant coprime polynomials with coefficients that are entire functions. Assume that $G(0,\dots,0)$ is not identically zero.

1. (a)

If $F(g_{1}^{k},\ldots,g_{n}^{k})/G(g_{1}^{k},\ldots,g_{n}^{k})$ are entire functions for infinitely many positive integers $k$; or 2. (b)

$g_{1},\ldots,g_{n}$* are entire functions without zeros and $F(g_{1},\ldots,g_{n})/G(g_{1},\ldots,g_{n})$ is an entire function,*

then there exists an index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$ such that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\in K_{\bf g}$.

5.2. Nevanlinna Theory with Moving Targets

Let ${\bf f}=(f_{0},\dots,f_{n})$ be a holomorphic map from $\mathbb{C}$ to $\mathbb{P}^{n}$ where $f_{0},f_{1},\ldots,f_{n}$ are holomorphic functions without a common zero. Let $a_{0},\cdots,a_{n}\in K_{\bf f}$, and let $L:=a_{0}X_{0}+\dots+a_{n}X_{n}$. Then $L$ defines a hyperplane $H$ in $\mathbb{P}^{n}(K_{\bf f})$. We note that $H(z)$ is the hyperplane determined by the linear form $L(z)=a_{0}(z)X_{0}+\dots+a_{n}(z)X_{n}$ for $z\in\mathbb{C}$ that is not a common zero of $a_{0},\cdots,a_{n}$, or a pole of any $a_{k}$, $0\leq k\leq n$. The definition of the Weil function, proximity function and counting function can be easily extended to moving hyperplanes. For example,

[TABLE]

where $h$ is a meromorphic function such that $ha_{0},\cdots,ha_{n}$ are entire functions without common zeros, $P=(x_{0},\ldots,x_{n})\in\mathbb{P}^{n}(\mathbb{C})$ and $z\in\mathbb{C}$. It’s clear that

[TABLE]

for $z\in\mathbb{C}$ which is not a a common zero of $a_{0},\cdots,a_{n}$, or a pole of any $a_{k}$, $0\leq k\leq n$. The first main theorem for a moving hyperplane $H$ can be stated as

[TABLE]

We will reformulate the second main theorem with moving targets stated in [16, Theorem A4.2.1] to suit our purpose. Let $a_{j0},\ldots,a_{jn}\in K_{\bf f}$, and let $L_{j}:=a_{j0}X_{0}+\dots+a_{jn}X_{n}$. Without loss of generality, we will normalize the linear forms $L_{j}$, $1\leq j\leq q$, such that for each $1\leq j\leq q$, there exists $0\leq j^{\prime}\leq n$ such that $a_{jj^{\prime}}=1$. Let $t$ be a positive integer and let $V(t)$ be the complex vector space spanned by the elements

[TABLE]

where the product and sum runs over $1\leq j\leq q$ and $0\leq k\leq n$. Let $1=b_{1},\cdots,b_{u}$ be a basis of $V(t)$ and $b_{1},\cdots,b_{w}$ a basis of $V(t+1)$. It’s clear that $u\leq w$. Moreover, we have [23, Lemma 6]

[TABLE]

The following formulation of the second main theorem with moving targets follows from the proof of [16, Theorem A4.2.1] by adding the Wronskian term when applying the second main theorem.

Theorem 5.4**.**

Let ${\bf f}=(f_{0},\ldots,f_{n}):\mathbb{C}\to\mathbb{P}^{n}(\mathbb{C})$ be a holomorphic curve where $f_{0},f_{1},\ldots,f_{n}$ are entire functions without common zero. Let $H_{j}$, $1\leq j\leq q$, be arbitrary (moving) hyperplanes given by $L_{j}:=a_{j0}X_{0}+\dots+a_{jn}X_{n}$ where $a_{j0},\cdots,a_{jn}\in K_{\bf f}$. Denote by $W$ the Wronskian of $\{hb_{m}f_{k}\,|\,1\leq m\leq w,\,0\leq k\leq n\}$, where $h$ is a meromorphic function such that $hb_{1},\ldots,hb_{w}$ are entire functions without common zero. If ${\bf f}$ is linearly non-degenerate over $K_{\bf f}$, then for any $\varepsilon>0$, we have the following inequality:

[TABLE]

where the maximum is taken over all subsets $J$ of $\{1,\dots,q\}$ such that $H_{j}(re^{i\theta})$, $j\in J$, are in general position.

The following two lemmas are moving targets versions of the Borel Lemma and Green’s theorem.

Lemma 5.5**.**

Let $f_{0},\ldots,f_{n}$ be entire functions with no zeros and $\mathbf{f}:=(f_{0},\ldots,f_{n})$ be a holomorphic map from $\mathbb{C}$ to $\mathbb{P}^{n}(\mathbb{C})$. Suppose that $f_{0},\ldots,f_{n}$ are linearly dependent over $K_{\bf f}$. Then for each $f_{i}$, there exists $j\neq i$ such that $f_{i}/f_{j}\in K_{\bf f}$.

The proof of this lemma is similar to the next one, therefore it is omitted.

Lemma 5.6**.**

Let $f_{0},\ldots,f_{n}$ be non-zero entire functions without a common zero and let ${\bf f}=(f_{0},\dots,f_{n})$ be a holomorphic map from $\mathbb{C}$ to $\mathbb{P}^{n}$. Assume that for an integer $k\geq n^{2}$ the following holds:

[TABLE]

where $a_{i}\neq 0\in K_{\bf f}$, $0\leq i\leq n$. Then for each $f_{i}$, there exists $j\neq i$ such that $(f_{i}/f_{j})^{k}\in K_{\bf f}$.

The proof of this lemma can be found in [7]. We include a slightly different proof for completeness.

Proof.

By reindexing the $f_{i}$, we may assume that $f_{0}^{k},f_{1}^{k},\cdots,f_{m-1}^{k}$, for some $1\leq m\leq n$, is a basis of the $K_{\bf f}$-vector space spanned by $f_{0}^{k},f_{1}^{k},\cdots,f_{n}^{k}$. It suffices to show that for $m\leq i\leq n$, there exists $0\leq j\leq m-1$ such that $(f_{i}/f_{j})^{k}\in K_{\bf f}$ as $f_{0}^{k},f_{1}^{k},\cdots,f_{m-1}^{k}$ are $K_{\bf f}$-linearly independent and the $f_{i}$ satisfy (39). Let $m\leq i\leq n$. After possibly reindexing, we obtain

[TABLE]

where $0\leq\ell\leq m-1$ and $\alpha_{0},\ldots,\alpha_{\ell}\in K_{\bf f}\setminus\{0\}$. If $\ell=0$, then $(f_{i}/f_{j})^{k}\in K_{\bf f}$ for some $0\leq j\leq m-1$. Suppose that $\ell\geq 1$. Let $\beta$ be an entire function such that $\tilde{f}_{0}=f_{0}/\beta,\ldots,\tilde{f}_{\ell}=f_{\ell}/\beta$ have no common zero, let $\tilde{f}_{i}=f_{i}/\beta$, and let

[TABLE]

Let $h$ be a meromorphic function such that $h\alpha_{0}\tilde{f}_{0}^{k},\ldots,h\alpha_{\ell}\tilde{f}_{\ell}^{k}$ are entire functions with no common zeros, and let

[TABLE]

be a holomorphic map from $\mathbb{C}$ to $\mathbb{P}^{\ell}(\mathbb{C})$. Then

[TABLE]

Similarly, by writing $\tilde{\mathbf{f}}_{k}=((h\alpha_{0})^{-1}h\alpha_{0}\tilde{f}_{0}^{k},\ldots,(h\alpha_{\ell})^{-1}h\alpha_{\ell}\tilde{f}_{\ell}^{k})$, we have

[TABLE]

Moreover, the map $\mathbf{F}_{k}$ is linearly non-degenerate over $\mathbb{C}$, since $f_{0}^{k},f_{1}^{k},\cdots,f_{\ell}^{k}$ are $K_{\bf f}$-linearly independent. Applying Theorem 2.7 to the map $\mathbf{F}_{k}$ with the coordinate hyperplanes $\{X_{j}=0\}$, $0\leq j\leq\ell$, and the diagonal hyperplane $\{X_{0}+\cdots+X_{\ell}=0\}$ of $\mathbb{P}^{\ell}(\mathbb{C})$, for any $\epsilon>0$ we have

[TABLE]

Together with (40), for all $\epsilon>0$ we have

[TABLE]

Hence,

[TABLE]

If $k\geq n^{2}\geq(\ell+1)^{2}>\ell^{2}+2\ell$, then this implies that $T_{\frac{f_{j}}{f_{0}}}(r)\leq T_{\tilde{\mathbf{f}}_{k}}(r)\leq_{\rm exc}{\rm o}(T_{{\mathbf{f}}}(r))$ for $1\leq j\leq\ell$, and hence $\frac{f_{j}}{f_{0}}\in K_{\bf f}$ for $1\leq j\leq\ell$, contradicting that $f_{0}^{k},f_{1}^{k},\cdots,f_{\ell}^{k}$ are linearly independent over $K_{\bf f}$. ∎

5.3. Key Theorem

The following fundamental result is the analogue of Theorem 3.1.

Theorem 5.7**.**

Let $g_{0},g_{1},\dots,g_{n}$ be entire functions without common zeros and let ${\bf g}=(g_{0},g_{1},\dots,g_{n})$. Let $F,G\in K_{\bf g}[x_{0},x_{1},\ldots,x_{n}]$ be coprime homogeneous polynomials of the same degree $d>0$. Let $I$ be the set of exponents ${\bf i}$ such that ${\bf x}^{\bf i}$ appears with a nonzero coefficient in either $F$ or $G$. Let $m\geq d$ be a positive integer. Suppose that the set $\{g_{0}^{i_{0}}\dots g_{n}^{i_{n}}:i_{0}+\cdots+i_{n}=m\}$ is linearly independent over $K_{\bf g}$. Then for any $\epsilon>0$, there exists a positive integer $L$ such that the following holds:

[TABLE]

where $c_{m,n,d}=2\binom{m+n-d}{n+1}-\binom{m+n-2d}{n+1}$, $M=2\binom{m+n-d}{n}-\binom{m+n-2d}{n}$, and $M^{\prime}$ is an integer of order $O(m^{n-2})$.

The basic ideas used to prove the theorem are similar to the ideas used in the proof of Theorem 3.1. We will make explicit the important differences in the moving target case and omit whatever is identical or obvious from the proof of Theorem 3.1.

Proof of Theorem 5.7.

Let $(F,G)$ be the ideal generated by $F$ and $G$ in $K_{\bf g}[x_{0},\cdots,x_{n}]$ and $(F,G)_{m}:=K_{\bf g}[x_{0},\cdots,x_{n}]_{m}\cap(F,G).$ We choose $\{\phi_{1},\ldots,\phi_{M}\}$ to be a basis of the $K_{\bf g}$-vector space $(F,G)_{m}$ consisting of elements of the form $F{\bf x}^{\bf i}$ and $G{\bf x}^{\bf j}$.

For each $z\in\mathbb{C}$, we construct a basis $B_{z}$ for $V_{m}=K_{\bf g}[x_{0},\ldots,x_{n}]_{m}/(F,G)_{m}$ as follows. For ${\bf i}=(i_{0},\cdots,i_{n})\in\mathbb{N}^{n+1}$, we let ${\bf g}(z)^{\bf i}=g_{0}(z)^{i_{0}}\cdots g_{n}(z)^{i_{n}}$. Choose a monomial ${\bf x}^{{\bf i}_{1}}$ so that $|{\bf g}(z)^{{\bf i}_{1}}|$ is minimal subject to the condition ${\bf x}^{{\bf i}_{1}}\notin(F,G).$ Suppose now that ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{j}}$ have been constructed. Then we let ${\bf x}^{{\bf i}_{j+1}}\in K_{\bf g}[x_{0},\ldots,x_{n}]_{m}$ be a monomial such that $|{\bf g}(z)^{{\bf i}_{j+1}}|$ is minimal subject to the condition that ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{j+1}}$ are linearly independent modulo $(F,G)_{m}$. In this way, we construct a basis of $V_{m}$ with monomial representatives ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{M^{\prime}}}$, where $M^{\prime}=\dim V_{m}$. Let $I_{z}=\{{\bf i}_{1},\ldots,{\bf i}_{M^{\prime}}\}$. Then for each ${\bf i}$, $|{\bf i}|=m$, we have

[TABLE]

for some choice of coefficient $b_{{\bf i},j,z}\in K_{\bf g}$. Then for each such ${\bf i}$ there is a linear form $L_{z,{\bf i}}$ over $K_{\bf g}$ such that

[TABLE]

We note that there are only finitely many choice of $b_{{\bf i},j,z}$ even as $z$ runs through all of $\mathbb{C}$, and $\{L_{z,{\bf i}}(\phi_{1},\ldots,\phi_{M})\,|\,|{\bf i}|=m,{\bf i}\notin I_{z}\}$ is a basis for $(F,G)_{m}$. From the definition of ${\bf x}^{{\bf i}_{1}},\ldots,{\bf x}^{{\bf i}_{M^{\prime}}}$, we have the key inequality

[TABLE]

where $\Phi({\bf g}(z))=(\phi_{1}({\bf g}(z)),\ldots,\phi_{M}({\bf g}(z)))$, $\|L_{z,{\bf i}}(z)\|=\max_{0\leq j\leq M^{\prime}}|b_{{\bf i},j,z}(z)|$, and $b_{{\bf i},0,z}=1$. The map $(\phi_{1}({\bf g}),\ldots,\phi_{M}({\bf g}))$ may not be a reduced presentation of $\Phi({\bf g})$. Let $h$ be a meromorphic function such that $F({\bf g})/h$ and $G({\bf g})/h$ are entire and have no common zeros, i.e.,

[TABLE]

Moreover, since $g_{i}$, $0\leq i\leq n$, are entire functions, the poles of $h$ comes from the poles of the coefficients of $F$ and $G$, and hence

[TABLE]

Let $\psi_{i}:=\phi_{i}({\bf g})/h$, $1\leq i\leq M$. By the choice of $\phi_{i}$, the function $\psi_{i}$ is entire for $1\leq i\leq M$. Furthermore, since $FX_{i}^{m-d},GX_{i}^{m-d}\in(F,G)_{m}$, $0\leq i\leq n$, and $g_{0},\ldots,g_{n}$ have no common zero, the functions $\psi_{i}$, $1\leq i\leq M$, have no common zero. Hence $\Psi:=(\psi_{1},\ldots,\psi_{M})$ is a reduced form of $\Phi({\bf g})$.

Let $V$ be the complex vector space spanned by $1$, the coefficients of $F$ and $G$, and all possible (finitely many) choices of $b_{{\bf i},j,z}$ in (41). Let $t$ be a large positive integer and let $V(t)$ be the finite-dimensional vector space spanned by products of $t$ elements in $V$.

Let $b_{1}=1,b_{2},\cdots,b_{u}$ be a basis of $V(t)$ and $b_{1},\cdots,b_{w}$ be a basis of $V(t+1)$. We will choose $t$ sufficiently large so that $\frac{w}{u}\leq 1+\frac{\epsilon}{2m}$. Denote by $W$ the Wronskian of $\{ab_{m}\psi_{k}\,|\,1\leq m\leq w,\,1\leq k\leq M\}$, where $a$ is an entire function such that $a=ab_{1},ab_{2},\cdots,ab_{w}$ are entire and have no common zeros. Applying Theorem 5.4 to the holomorphic map $\Phi({\bf g})$ with the reduced form $\Psi=(\psi_{1},\ldots,\psi_{M})$ and the set of linear forms $\{L_{z,{\bf i}}\,|\,|{\bf i}|=m,{\bf i}\notin I_{z}\}$, we have the following inequality:

[TABLE]

where

[TABLE]

if $z=re^{i\theta}$ is not a pole of any coefficient of $L_{z,{\bf i}}$. We note that such $z$ is in a discrete subset of $\mathbb{C}$, and hence its radius $r\in(0,\infty)$ is in a set of finite Lebesgue measure. In the following computation, we will only consider $z$ which is not a pole of any $b_{{\bf i},j,z}$ from (41). Next, we will derive a lower bound for the first term of the left hand side of (45). By the definition of the characteristic function, Lemma 2.1, (43) and (44), we have

[TABLE]

On the other hand, since $\phi_{i}=\sum_{I}\alpha_{I}x^{I}\in K_{\bf g}[x_{0},\ldots,x_{n}]_{m}$,

[TABLE]

and hence

[TABLE]

In conclusion, we have

[TABLE]

Now, we use the key inequality (42), the estimates in (12) and the two equations following it to derive

[TABLE]

for $z\in\mathbb{C}$ that is not a pole of any $b_{{\bf i},j,z}$ from (41). By Lemma 2.1, Jensen’s formula, the integration of (48) from 0 to $2\pi$ over $d\theta$ gives

[TABLE]

Together with (45) and (47), we have

[TABLE]

Since $F$ and $G$ are coprime, the ideal $(F,G)$ defines a closed subset of $\mathbb{A}^{n}$ of codimension at least 2. As is well-known from the theory of Hilbert functions and Hilbert polynomials, this implies that $M^{\prime}=O(m^{n-2})$.

It then suffices to show that there exists a large integer $L$ (to be determined later) such that

[TABLE]

The above inequality can be deduced from the inequality

[TABLE]

for $z\in\mathbb{C}$. The inequality holds trivially if $v_{z}^{+}(g_{i})\leq L$ for each $1\leq i\leq n$. Therefore, we only need to consider the case where $v_{z}^{+}(g_{i})>L$ for some $1\leq i\leq n$.

For $z\in\mathbb{C}$, we define a monomial ordering $>_{{\bf g}(z)}$ on $A=K_{\bf g}[x_{0},\cdots,x_{n}]$ using the weight vector ${\bf u}=(v_{z}(g_{0}),\ldots,v_{z}(g_{n}))$, where we set $g_{0}=1$. Let

[TABLE]

where $\{F_{1},F_{2}\}=\{F,G\}$ and ${\rm TM}_{{\bf g}(z)}(F_{2})\leq{\rm TM}_{{\bf g}(z)}(F_{1})$. By Lemma 2.11, $B=(B_{1}\setminus B_{1}^{\prime})\cup B_{2}$ is a basis for $(F,G)_{m}$. Write $B=\{\beta_{1},\ldots,\beta_{M}\}$, which depends on $z$, although there are only a finite number of choices of such a basis. Let $\eta_{j}=\beta_{j}({\bf g})/h$ (note that the coefficients of the $\beta_{j}$ come from the coefficients of $F$ and $G$). From the definition of $>_{{\bf g}(z)}$ and $F_{2}$, it follows that

[TABLE]

where $I$ is the set of exponents ${\bf i}$ such that ${\bf x}^{\bf i}$ appears with a nonzero coefficient in either $F$ or $G$. Then similar to (21), by the second part of Lemma 2.11, we have for each $z\in\mathbb{C}$,

[TABLE]

On the other hand, from the definition of the $\phi_{i}$ and $\beta_{i}$, we see that $\eta_{j}$ is a $V$-linear combination of $\psi_{k}$, $1\leq k\leq M$. Therefore, $ab_{m}\eta_{j}$, $1\leq m\leq u$, $1\leq j\leq M$, is a $\mathbb{C}$-linear combination of $\{ab_{m}\psi_{k}\,|\,1\leq m\leq w,\,1\leq k\leq M\}$. As $W$ is the Wronskian of $\{ab_{m}\psi_{k}\,|\,1\leq m\leq w,\,1\leq k\leq M\}$, where $a$ is an entire function such that $a=ab_{1},ab_{2},\cdots,ab_{w}$ are entire and have no common zeros, from the basic properties of Wronskians we have

[TABLE]

Combining (55) and (56), we obtain that

[TABLE]

Let $L=\frac{1}{2u}Mw(Mw-1)c_{m,n,d}^{-1}$. The assumption that $v_{z}^{+}(g_{i})>L$ for some $0\leq i\leq n$ implies that

[TABLE]

Therefore, the inequality (54) can be deduced from (57) and (58). ∎

5.4. Proof of the main theorems

The following theorem is the moving target version of Theorem 4.3. Denote by $R_{\bf g}\subset K_{\bf g}$, the subring of entire functions

Theorem 5.8**.**

Let $g_{1},\ldots,g_{n}$ be entire functions without common zeros and ${\bf g}=(1,g_{1},\dots,g_{n})$. Let $G\in R_{\bf g}[x_{1},\ldots,x_{n}]$ be a nonconstant polynomial of degree $d$ such that $G(0,\ldots,0)$ is not identically zero. Assume that $g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\notin K_{\bf g}$ for any index set $(i_{1},\dots,i_{n})\in\mathbb{Z}^{n}\setminus\{(0,\dots,0)\}$. For all $\epsilon>0$,

1. (a)

there exists a positive integer $k_{0}$ such that for all $k\geq k_{0}$,

[TABLE] 2. (b)

if in addition each $g_{i}$ has no zero, $1\leq i\leq n$, then

[TABLE]

Proof.

By Lemma 5.6, the set $\{{\bf g}^{k{\bf i}}:=g_{1}^{ki_{1}}\cdots g_{n}^{ki_{n}}\,|\,{\bf i}=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n},|{\bf i}|\leq d\}$ is $K_{\bf g}$-linearly independent for $k\geq\binom{d+n}{n}^{2}$. Since $G(0,\ldots,0)\neq 0$, by arranging the index set in some order, we may write

[TABLE]

where $a_{{\bf i}(j)}\in R_{\bf g}\setminus\{0\}$ for $0\leq j\leq\ell$. Then we have

[TABLE]

Let $h$ be an entire function such that $h^{-1}a_{{\bf i}(0)},h^{-1}a_{{\bf i}(1)}{\bf g}^{k{\bf i}(1)},\ldots,h^{-1}a_{{\bf i}(\ell)}{\bf g}^{k{\bf i}(\ell)}$ have no common zero. Hence,

[TABLE]

We apply Theorem 2.7, Cartan’s truncated second main theorem, to the holomorphic map

[TABLE]

associated with the above expression ($k\geq\binom{d+n}{n}^{2}$), and with the set of coordinate hyperplanes of $\mathbb{P}^{\ell}$ and the diagonal hyperplane $\sum_{i=0}^{\ell}X_{j}$. Then for any $\epsilon>0$, we have

[TABLE]

We note that

[TABLE]

Then for $k>\max\{\frac{2\ell^{2}}{\epsilon},\binom{d+n}{n}^{2}\}$,

[TABLE]

Since $G(g_{1}^{k},\ldots,g_{n}^{k})$ is an entire function,

[TABLE]

Consequently,

[TABLE]

When the $g_{i}$ are entire functions without zeros, we may assume that the set $\{{\bf g}^{{\bf i}}:=g_{1}^{i_{1}}\cdots g_{n}^{i_{n}}\,|\,{\bf i}=(i_{1},\ldots,i_{n})\in\mathbb{N}^{n},|{\bf i}|\leq d\}$ is linearly independent over $K_{\bf g}$ by Lemma 5.5. Then we can repeat the previous argument for $k=1$ with the additional condition $N_{g_{i}}(0,r)=O(1)$, $1\leq i\leq n$, to conclude the proof. ∎

The proofs of Theorem 5.1, Theorem 5.2 and Corollary 5.3 are similar to their constant analogues as the necessary tools have been developed. Therefore, we will not repeat the proofs here.