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| 研究生： |
江泰緯 |
|---|---|
| 論文名稱： |
熱帶直線建構二次及三次熱帶曲線之研究 Constructing Tropical Curves of Degree Two and Three with Tropical Lines |
| 指導教授： | 蔡炎龍 |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2013 |
| 畢業學年度： | 101 |
| 語文別： | 英文 |
| 論文頁數： | 55 |
| 中文關鍵詞： | 熱帶多項式 、熱帶曲線 、熱帶直線 |
| 外文關鍵詞： | Tropical Polynomial, Tropical Curve, Tropical Line |
| 相關次數： | 點閱：78 下載：26 |
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在這篇論文裡， 我們找到了一個方法來反推出對應到某個熱帶曲線的熱帶 多項式。在給定一個二次或三次的熱帶曲線之後， 我們利用熱帶直線來找出 此熱帶曲線的多項式。再來， 若給定一個二次或三次的牛頓細分(Newton subdivision) ， 我們也能找出能對應到它的熱帶多項式。
In this thesis, we develop an algorithm to recover tropical polynomials from plane tropical curves of degree two and three. We use tropical lines to approach a given tropical curve. Furthermore, we also give another algorithm to recover tropical polynomials from a (maximal) Newton subdivision of degree two and three.
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 Introduction 1
2 Tropical Algebraic Geometry 3
2.1 Tropical polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Tropical curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Tropical factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Recovering Tropical Polynomials from Tropical curves 19
3.1 Tropical curves of degree two . . . . . . . . . . . . . . . . . . . . . 19
3.2 Tropical curves of degree three . . . . . . . . . . . . . . . . . . . . . 29
4 Recovering Tropical Polynomials from Newton Subdivisions 36
4.1 Newton subdivisions of degree two . . . . . . . . . . . . . . . . . . 36
4.2 Newton subdivisions of degree three . . . . . . . . . . . . . . . . . . 38
A All types of maximal Newton subdivisions of degree three 51
Bibliography 55
[1] Andreas Gathmann. Tropical algebraic geometry. Jahresber. Deutsch. Math.- Verein., 108(1):3–32, 2006.
[2] Nathan Grigg. Factorization of tropical polynomials in one and several variables. Honor’s thesis, Brigham Young University, 2007.
[3] Grigory Mikhalkin. Counting curves via lattice paths in polygons. C. R. Math. Acad. Sci. Paris, 336(8):629–634, 2003.
[4] Grigory Mikhalkin. Enumerative tropical algebraic geometry in R2. J. Amer. Math. Soc., 18(2):313–377, 2005.
[5] Jürgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. In Idempotent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317. Amer. Math. Soc., Providence, RI, 2005.
[6] Imre Simon. Recognizable sets with multiplicities in the tropical semiring. In Michal Chytil, Ladislav Janiga, and Václav Koubek, editors, MFCS, volume 324 of Lecture Notes in Computer Science, pages 107–120. Springer, 1988.
[7] David Speyer. Tropical geometry. PhD thesis, UC Berkeley, 2005.
[8] Yen-Lung Tsai. Working with tropical meromorphic functions of one variable. Taiwanese J. Math., 16(2):691–712, 2012.
