國立清華大學 數學系

Abstract 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 Hussein and Ru's theorem 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 $\mathcal N=\{n\in\mathbb{N} |F(n)/G(n)\text{ is an entire function}\}$ 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.