對於互素的遞歸數列$\{{\bf F}(n)\}_{n\in\mathbb N}$和$\{{\bf G}(n)\}_{n\in\mathbb N}$，我們知道僅有有限多個正整數$n$使得${\bf F}(n)$被${\bf G}(n)$整除。本文中我們將研究該整除問題在複變函數中的類比。

首先，假設$f$和$g$是乘法獨立的整函數。我們希望研究是否存在無窮多個正整數$n$使得$f^n-1$會被$g^n-1$整除。這一問題需要對$f^n-1$和$g^n-1$公因子上界進行一個合適的估計。為了得到這一估計，我們需要把\cite{hussein2018general}中的主要結果改進成truncated version。然後我們會把定理中的variety取為$\PP^1\times\PP^1$的blow-up，並且計算該定理中相應的常數。

接下來，我們會推廣上述整除問題。準確地說，我們將研究$F(n)=a_0+a_1f_1^n+\cdots+a_lf_l^n$是否會被$ G(n)=b_0+b_1g_1^n+\cdots+b_mg_m^n$整除，其中，$f_i$和$g_j$都是非常數的整函數，而除了$a_0$可能為零以外，$a_i$和$b_j$都是非零常數。我們的結論是如果對於不全是零的整數組$(i_1,\dots,i_l,j_1,\dots,j_m)$，$f_1^{i_1} \cdots f_l^{i_l}g_1^{j_1}\dots g_m^{j_m}$都不是整數，那麼$\mathcal N=\{n\in\NN |F(n)/G(n)\text{ 仍然是一個整函數}\} $是一個有限集合。我們接下來考慮更一般的情況。假設$a_i$和$b_j$相對於$(g_1,\dots,g_m)$緩慢增長的，那麼我們有類似的結論。

Let $\{{\bf F}(n)\}_{n\in\mathbb N}$ and $\{{\bf G}(n)\}_{n\in\mathbb N}$ be linear recurrence sequences. It is a well-known diophantine problem to decide the finiteness of the set $\mathcal N$ of natural numbers such that their ratio ${\bf F}(n)/{\bf G}(n)$ is an integer. In this thesis, we study an analogue of such a quotient problem in the complex situation.

First, let $f$ and $g$ be entire functions which are multiplicatively independent. We want to determine whether $f^n-1$ is divisible by $g^n-1$ for infinitely many $n$. This is an application of the GCD estimate of $f^n-1$ and $g^n-1$, i.e. the Nevanlinna counting function for the common zeros of these two sequences of functions. For this estimate, we need to formulate a truncated Nevanlinna second main theorem for effective divisors by modifying a theorem in \cite{hussein2018general} and explicitly computing the constants involved for a blow-up of $ \mathbb{P}^1\times \mathbb{P}^1$ along a point.

Next, we generalize the quotient problem to a multi-variable version. Precisely, we are concerned with the divisibility problem (in the sense of complex entire functions) for two sequences $F(n)=a_0+a_1f_1^n+\cdots+a_lf_l^n$ and $ G(n)=b_0+b_1g_1^n+\cdots+b_mg_m^n$, where the $f_i$ and $g_j$ are nonconstant entire functions and the $a_i$ and $b_j$ are non-zero constants except that $a_0$ can be zero. We will show that the set $\mathcal N=\{n\in\NN |F(n)/G(n)\text{ is an entire function}\}$ is finite under the assumption that $f_1^{i_1} \cdots f_l^{i_l}g_1^{j_1}\dots g_m^{j_m}$ is not constant for any non-trivial index set $(i_1,\dots,i_l,j_1,\dots,j_m)\in\mathbb Z^{l+m}$. We also consider the generalization of this problem in which we allow $a_i$ and $b_j$ to be slow growth entire functions with respect to $(g_1,\dots,g_m)$ by modifying the second main theorem with moving targets to a truncated version.

1 Introduction 1
1.1 The Quotient Problem for Linear Recurrences . . . . . . . . . . . . . . 2
1.2 The GCD Problem of Linear Recurrences . . . . . . . . . . . . . . . . 3
1.3 Main Results on the Quotients of Entire Functions . . . . . . . . . . . . 6
1.3.1 Divisibility of fn 􀀀 1 and gn 􀀀 1 . . . . . . . . . . . . . . . . 7
1.3.2 Quotient Problem for Entire Functions . . . . . . . . . . . . . . 8
1.3.3 Quotient Problem for Entire Functions with Moving Targets . . 9
2 Preliminaries 11
2.1 Nevanlinna Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Roth’s Theorem and the Subspace Theorem . . . . . . . . . . . . . . . 18
2.3 Vojta’s Dictionary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Further Results in Nevanlinna Theory 23
3.1 Second Main Theorem for Divisors in Subgeneral Position . . . . . . . 23
3.2 Nevanlinna theory with moving targets . . . . . . . . . . . . . . . . . . 27
3.2.1 A Second Main Theorem with Moving Targets . . . . . . . . . 27
3.2.2 Borel’s lemma and Green’s theorem with moving targets . . . . 33
4 The Asymptotic GCD for Entire Functions fn 􀀀 1 and gn 􀀀 1 37
4.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 The Proof of Theorem 1.11 . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Quotient Problems for Entire Functions 45
5.1 The Quotient for fn 􀀀 1 and gn 􀀀 1 . . . . . . . . . . . . . . . . . . . 45
5.1.1 Unit Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.2 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Proof of Theorem 1.12 . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Proof of Corollary 1.14 and Theorem 1.13 . . . . . . . . . . . . . . . . 58
A Exponential Polynomials 66
Bibliography 68

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