熱帶幾何(tropical geometry)在近年來引起數學家的注意，因為它可以簡化許多數學難題。本篇論文主要在探討單變數熱帶多項式(single variable tropical polynomial)的因式分解。對於每個熱帶多項式，我們都可以定義其對應的最大係數熱帶多項式(largest-coefficient tropical polynomial )，而且此最大係數熱帶多項式可以因式分解為線性乘積。根據此結果，熱帶代數基本定理(Fundamental Theorem of Tropical Algebra)即成立。此外，可將單變數熱帶多項式因式分解的許多概念延伸至多變數的情形。 / Tropical geometry draw much attention recent years for it simplifies many difficult classical mathematics problems. The thesis mainly discuss factorization of single variable tropical polynomials. For every tropical polynomial, we define the corresponding largest-coefficient tropical polynomial. We show that each largest-coefficient tropical polynomial can be factorized into a product of linear terms. As a result, the Fundamental Theorem of Tropical Algebra holds. Furthermore, we observe that many notions of factorization of single variable tropical polynomial can be extended to multivariate cases.

在這篇論文中,我們定義了熱帶導數和熱帶反導數.當我們對兩個相同的熱帶多項式求導數時,可能會得到不同的函數.為了克服此困難,我們限制在最大係數多項式下才求導數.熱帶導數的定義與古典導數相當不同.特別的是,我們有d/dxan⊙x^(⊙n)= an⊙x⊙n-1.將它線性化,我們得到d/dx[an⊙x^(⊙n)⊕an-1⊙x⊙n-1 ⊕…. a1⊙x⊕a0] = an⊙x⊙n-1 ⊕an-1⊙x⊙n-2⊕…⊕a1.我們將會解釋為什麼使用這種定義.導數對了解熱帶幾何很有幫助,它也引出了一些與古典導數相似的資訊.最後,我們討論如何定義及求熱帶多項式的熱帶反導數 / In this thesis, we define the tropical derivatives and anti-derivatives. When we differ- entiate two identical tropical polynomials, we might get two different functions. In order to overcome the diffculties, we restrict the polynomials to largest coeffcient polynomials to avoid unpredictable results when taking derivatives. The definitiion of the tropical derivatives is quite diffrent from the definition of classical derivatives. In particular, we have d/dxan⊙x^(⊙n)= an⊙x⊙n-1 . To extend it linearly, we obtain d/dx[an⊙x^(⊙n)⊕ a n-1⊙x⊙n-1 ⊕…. a1⊙x⊕a0] = an⊙x⊙n-1 ⊕a n-1⊙x⊙n-2⊕…⊕a1. We will explain why we use this kind of definition. The derivatives are helpful in understanding more about tropical geometry, and it carries out some information similar to classical derivatives. Finally, we discuss how to define and find tropical anti-derivatives for tropical polynomials. Keywords : Tropical derivatives, tropical anti-derivatives, tropical polynomials.

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.