In this paper, we study the problem of uniqueness on meromorphic functions involving in differential polynomials and obtain some results which extend and improve the theorems of M. Fang and W. Hong et al.

在這篇論文裡，我們探討 $\mathcal{A}$ 類半純函數的值分佈基本理論。我們證明了每一個 $\mathcal{A}$ 類半純函數最多有兩個重值，而這個結果是最佳的情形。進而，我們證明若一個 $\mathcal{A}$ 類半純函數 $f$ 與其導數 $f^{(k)}$ 共非零的複數值，則 $f\equiv f^{(k)}$。 / In this thesis, we study the basic theory of value distribution of meromorphic function of class $\mathcal{A}$. We prove that every meromorphic function of class $\mathcal{A}$ has at most two multiple values and the result is sharp. Also, we prove that if a meromorphic function $f$ of class $\mathcal{A}$ and its derivative $f^{(k)}$ share a non-zero complex value, then $f\equiv f^{(k)}$.

Haymen猜測:對任意的超越半純函數 f(z)，f'(z)f(z)^n 取所有值無窮多次，其中至多只有一個例外值。這個著名的猜測，大部分的情形已被證明是正確的。另外，Hayman 證明 f'(z)-af(z)^n 取所有有限值無窮多次 ，其中 a 為一複數且 n≧5 的正整數。在本篇論文裡，我們將探討以小函數為係數的半純函數微分多項式之值分佈問題。並將Hayman的結果推廣至 f^{k}(z)f(z)^n 與 f^{k}(z)-af(z)^n 的情形。同時，我們也證明一些 A類半純函數與其導數的值分佈結果。 / A famous conjecture of Hayman says that if f(z) is a transcendental meromorphic function, then f'(z)f(z)^n assumes all finite values except possibly zero infinitely often. The conjecture was solved in most cases. Another result of Hayman says that f'(z)-af(z)^n, where n≧5 and a is a complex number, assumes all finite values infinitely often. In this thesis, we will study the value distribution of some differential polynomial in a meromorphic function with small functions as coefficents. In fact, we will generalize Hayman's results to the cases f^(k)(z)f(z)^n and f^(k)(z)-af(z)^n. Also, the value distribution of meromorphic functions of class A with their derivatives are obtained.

Cette thèse porte sur des propriétés arithmétiques des fonctions méromorphes et transcendantes d'une variable. Dans le chapitre 3, nous définissons des mesures de transcendance pour les fonctions holomorphes et méromorphes sur un domaine régulier de C puis nous majorons ces mesures en fonction de la distribution des petites valeurs de la fonction étudiée.Grâce aux théories de Nevanlinna et d'Ahlfors, nous étudions dans le chapitre 4 la distribution des petites valeurs de certaines classes de fonctions méromorphes sur D ou C afin d'obtenir pour celles-ci des majorations explicites de leurs mesures de transcendance. L'application principale de ce travail est l'obtention de nouveaux lemmes de zéros polynomiaux pour de grandes familles de fonctions méromorphes et en particulier pour les fonctions de Weierstrass et les fonctions fuchsiennes. Dans le chapitre 5, nous montrons que ces lemmes de zéros polynomiaux conduisent à des bornes logarithmiques du nombre de points algébriques de degré et hauteur bornée contenus dans les graphes des fonctions étudiées. / This PhD thesis is about some arithmetic properties of meromorphic functions of one variable.In chapter 3, we define the transcendental measures for holomorphic and meromorphic functions on a regular domain of C, then we obtain upper bounds of these measures in terms of the distribution of small values of the function.Thanks to the Nevanlinna and Ahlfors theories, we study in chapter 4 the distribution of small values of some classes of meromorphic functions on D or C in order to obtain explicit upper bounds of transcendental measures.The main application of this work is the demonstration of new polynomial zero estimates for large classes of meromorphic functions, in particular for Weierstrass functions and fuchsian functions.In chapter 5, we prove that polynomial zero estimates lead to logarithmic bounds of the number of algebraic points of bounded degree and height contained in the graph of the function.

在這篇論文裡，我們利用值分佈的理論來探討半純函數的共值與唯一性的問題，本文包含了以下的結果：將Jank與Terglane有關三個A類中的半純函數唯一性的結果推廣到任意q個半純函數的情形；證明了C. C. Yang的一個猜測；建構了一類半純函數恰有兩個虧值，而且算出它們的虧格；將 Nevanlinna 五個值的定理推廣至兩個半純函數部分共值的情形；探討純函數 與其導數的共值問題；最後，證明了兩個半純函數共四個值且重數皆不同的定 理。 / In this thesis, we study the sharing value problems and the uniqueness problems of meromorphic functions in the theory of value distribution. In fact, this thesis contains the following results: We generalize a unicity condition of three meromorphic functions given by Jank and Terglane in class A to the case of arbitrary q meromorphic functoins. An elementary proof of a conjecture of C. C. Yang is provided. We construct a class of meromorphic functions with exact two deficient values and their deficiencies are explicitly computed. We generalize the Nevanlinna's five-value theorem to the cases that two meromorphic functions partially share either five or more values, or five or more small functions. In each case, we formulate a way to measure how far these two meromorphic functions are from sharing either values or small functions, and use this measurement to prove a uniqueness theorem. Also, we prove some uniqueness theorems on entire functions that share a pair of values (a,-a) with their derivatives, which are reformulations of some important results about uniqueness of entire functions that share values with their derivatives. Finally, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ share four distinct values a_1, a_2, a_3, a_4 DM such that each a_i-point is either a (p,q)-fold or (q,p)-fold point of f and g, then (p,q) is either (1,2) or (1,3) and f, g are in some particular forms.

在這篇論文中，我們將利用值分佈的理論來探討下列函數方程解的存在性與其性質： \[\sum_{j=1}^pa_j(z)f_j(z)^{k_j}=1,\] 其中 $a_1(z),\cdots ,a_p(z)$ 為半純函數。對某些特殊方程，除了文獻裡已知的結果外，我們亦提供其它的例子。一般而言，我們探討解存在的必要條件。另外，我們證明了某一類半純函數之零點與極點之分佈的結果。 / In this thesis, we use the theory of value distribution to study the existence of solution of the following functional equation: \[\sum_{j=1}^pa_j(z)f_j(z)^{k_j}=1,\] where $a_1(z),\cdots ,a_p(z)$ are meromorphic functions. For some special case, new and old examples of the solutions are given. For the general case, a necessary condition for the existence of solution is considered. Moreover, we obtain a result on the distribution of zeros and poles of a class of meromorphic functions.

在這篇論文中，一個熱帶亞純函數(tropical meromorphic function)若給定有限個零根(roots)與極點(poles)還有它們的重數(multiplicities)，我們證明了這個熱帶亞純函數的存在與唯一性。另外，一個熱帶週期函數(tropical periodic function)若給定一個週期區間內的有限個零根與極點還有它們的重數，我們也找到了這個熱帶週期函數的一個簡單表達式。接著，給定一個一階線性差分方程(first-order linear difference equation)，我們討論了各種情況下的所有熱帶亞純函數解的表達式。最後，對於連續函數我們提供了一個它的熱帶近似函數，希望對於解差分方程的熱帶亞純函數解時能有所助益。 / In this thesis, we find the formula of tropical meromorphic function by giving finite number of roots and poles (with multiplicities). We also find a simple formula for tropical periodic function by giving finite number of roots and poles (with multiplicities) during a period [0,T). We then discuss all cases of the tropical meromorphic solution functions of first-order linear difference equation. At last, we provide a tropical approximated function of a given continuous function. We hope it is helpful in solving the tropical meromorphic solution functions of a given difference equation.