在這篇論文裡,我們主要在探討熱帶幾何的基本性質並研究熱帶幾何近來的發展,特別是Mikhalkin計算曲線個數的方法.最後,我們簡短的討論熱帶幾何在三維的一些情況. / In this thesis, we study the properties of tropical geometry and survey the recent development of tropical geometry, especially Mikhalkin's method of counting curves. We also briefly study tropical geometry in three-dimensional case.

本篇論文主要是在討論熱帶曲線上，離散型的因子應具備有哪些性質，而在討論 熱帶曲線上因子的性質時，我們會發現到致的計算，在低次熱帶曲線上，或是 因子本身的係數總和高時，並不是太困難。但再高次的熱帶曲線上，致的計算 就成線的極其繁雜，因此我們想要藉由在熱帶曲線上離散型的來幫助我們計算。 最後我們可以得到的結論是，若C是一平滑的熱帶曲線，則C上因子的致， 將會受到C的虧格所限制，大幅的簡化了我們在離散型熱帶曲線因子致的 計算。

此篇論文我們主要是探討熱帶幾何下，凸集(convex set)以及圓錐(cone)的生成元素(generator)個數。在第二章中我們對一些基本環境及運算工具做介紹，例如：熱帶半環(tropical semiring)為度量空間(metric space)、極限值的運算性質等等，在第三章中我們探討回收錐(recession cone)、凸集及圓錐的性質，其中包含三者之間的關係，而在第四章中我們探討閉圓錐(closed cone)、緊緻凸集(compact convex set)、閉凸集(closed convex set)三者的生成元素個數，並以實例說明此性質，最後我們將推論出一個方法來找出在二維的熱帶空間底下的有限生成圓錐之生成元素。 / In this thesis, I will discuss the generators of cone and convex set in tropical geometry. In Chapter 2, basic environment in tropical geometry and arithmetic tools are introduced here, such as how to find the limit value in tropical geometry or deciding if tropical semiring is a metric space, etc. In Chapter 3, I explore the properties of cone, convex and recession cone, inclusive of the relations of one another. In Chapter 4, the generators of a closed cone, a compact convex set, a closed convex set are provided with illustrations to present the properties. It will finally lead to a method to find the generators of the finitely generated cone in two dimesion space.

熱帶曲線(tropical curves) 是定義在熱帶半環(tropical semiring) 上的代數曲線。熱帶曲線是古典代數曲線經由某些賦值(valuation) 的映像，所以許多重要的代數曲線性質也同樣發生在熱帶曲線上。本篇論文我們試著將熱帶曲線圖形化。 首先，我們根據熱帶曲線的理論發展出幾個繪出熱帶曲線的演算法。再者，我們以電腦程式語言Python 去實現這些算演算法。我們發展的是跨平台的程式碼，可以在Linux, Mac OS X, Windows 等作業系統執行。 / Tropical curves are algebraic curves dened over the tropical semiring. They are the images of classical algebraic curves under some valuation maps, so reect many important properties of classical algebraic curves. In this thesis, we try to visualize tropical curves. We study the theory of tropical curves and develop several algorithms to draw the graphs of tropical curves. Furthermore, we implement these algorithms in Python programming language. These codes are cross-platform, running on Linux, Mac OS X, and Windows.

這一篇論文是說明矩陣中的元素如果滿足熱帶格雷斯曼就可以將它用樹型圖表示。第二章簡述熱帶數學的基本運算和介紹超曲面，作為熱帶格雷斯曼的先備知識。第三章介紹演化樹，在第二節中用兩種方法重建演化樹，第一種方法UPGMA是要在特定的條件下所建造的演化樹才會正確，第二種方法是利用矩陣中兩個物種間的距離假設的樹型配置圖，解聯立方程組得到各段距離，第三節中先介紹代數幾何中的格雷斯曼再聯結到熱帶幾何中的格雷斯曼。第四章是將第二種方法寫成Python程式，輸入矩陣中的元素，如果滿足熱帶格雷斯曼，就可以顯示它可能的樹型圖以及算出距離。

在這篇論文中，我們從K energy的角度探討緊緻法諾超平面上的K穩定性。首先，我們給K energy一個較明確的型式，接著再透過分析的手法求解其導函數。後續，我們引進熱帶幾何的結構來重新分析主要的結果，最後給一些法諾超平面的實例，驗證我們所得到的公式。 / In this thesis, we analyze K stability on compact Fano hypersurfaces from K energy. We first represent the K energy into an explicitly formula. Then we compute the derivative by using some analytic techniques. Furthermore, we introduce some structures of tropical geometry to analyze the main result. Finally, we give some examples of compact Fano hypersurface to test and verify the formula we get.

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.

本篇文章主要研究熱帶幾何之圓錐曲線，即二元二次多項式``根''的圖形。在文章中，我們以二元二次多項式係數關係做曲線的分類，歸納出20種熱帶圓錐曲線圖形，並證明此為完整的熱帶圓錐曲線之分類。然後，我們進一步討論如何調整二元二次多項式係數使圖形平移。最後，提出以熱帶直線輔助熱帶圓錐曲線快速作圖的方式。 / The purpose of the present study is to investigate conics -the graphs of the ``roots'' of quadratic polynomial- in tropical geometry. First, we induct and classify twenty types of tropical conics based on the relation between the coefficients and roots in quadratic polynomial. Second, evidences are provided to prove the classification thorough and intact. Then, we further discuss how to modify the quadratic polynomial in order to translate the graphs. Finally, suggestion about how to use tropical line to assist the graphing of tropical conics more efficiently is provided.

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.