摘要
對於互素的遞歸數列${{ f F}(n)}_{ninmathbb N}$和${{ f G}(n)}_{ninmathbb N}$，我們知道僅有有限多個正整數$n$使得${ f F}(n)$被${ f G}(n)$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$PP^1 imesPP^1$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={ninNN |F(n)/G(n) ext{ 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。
並列摘要
Let ${{ f F}(n)}_{ninmathbb N}$ and ${{ f G}(n)}_{ninmathbb N}$ 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 ${ f F}(n)/{ f G}(n)$ 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 imes 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 $mathcal N={ninNN |F(n)/G(n) ext{ is an entire function}}$ 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 $(i_1,dots,i_l,j_1,dots,j_m)inmathbb Z^{l+m}$. 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.