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| Title: | 熱帶橢圓曲線之研究
On Tropical Elliptic Curves |
| Authors: | 黃明怡
Huang, Ming Yi |
| Contributors: | 蔡炎龍
Tsai, Yen lung
黃明怡
Huang, Ming Yi |
| Keywords: | 熱帶幾何
橢圓曲線
因子理論
Picard 群
tropical geometry
elliptic curve
divisor theory
Picard group |
| Date: | 2014 |
| Issue Date: | 2015-02-03 10:24:54 (UTC+8) |
| Abstract: | 在數學許多分枝中, 橢圓曲線都是一個非常重要的主題, 例如在數論及代數幾何中 等等。本篇論文主要是研究熱帶幾何中的橢圓曲線。首先, 我們先討論什麼是熱帶橢 圓曲線的合理定義。接著我們研究熱帶橢圓曲線上的因子理論。如同古典的情況"所有"在熱帶橢圓曲線上的點和該曲線的 Picard 群是一一對應的。更進一步的說, 我們 還可在熱帶橢圓曲線上給一個群的結構。最後, 我們指出幾個未來可能的研究方向。
Elliptic curves has been important studying objects in many mathematics areas, such as number theory and algebraic geometry. In this thesis, we study tropical analogue of elliptic curves. We first discuss what is a reasonable way to define tropical elliptic curves. Then, we survey divisor theory on tropical elliptic curves. Like in classical elliptic curves, all “points” in a tropical elliptic curves are one-to-one corresponding to the Picard group of that elliptic curves. Moreover one can de- fine group structures on any tropical elliptic curves. Finally, we give some possible projects for future studies. |
| Reference: | [1] Omid Amini. Reduced divisors and embeddings of tropical curves, 2010.
[2] Yang An, Matthew Baker, Greg Kuperberg, and Farbod Shokrieh. Canonical represen- tatives for divisor classes on tropical curves and the matrix-tree theorem. Forum Math. Sigma, 2:e24, 25, 2014.
[3] Magnus Dehli Vigeland. The group law on a tropical elliptic curve. Math. Scand., 104(2): 188–204, 2009.
[4] Andreas Gathmann. Tropical algebraic geometry. Jahresber. Deutsch. Math.-Verein., 108(1):3–32, 2006.
[5] Jan Hladký, Daniel Král’, and Serguei Norine. Rank of divisors on tropical curves, 2010.
[6] San Ling, Huaxiong Wang, and Chaoping Xing. Algebraic curves in cryptography. Dis- crete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, 2013.
[7] Alfred Menezes. Elliptic curve public key cryptosystems. The Kluwer International Series in Engineering and Computer Science, 234. Kluwer Academic Publishers, Boston, MA, 1993. With a foreword by Neal Koblitz, Communications and Information Theory.
[8] Grigory Mikhalkin. Tropical geometry and its applications. In International Congress of Mathematicians. Vol. II, pages 827–852. Eur. Math. Soc., Zürich, 2006.
[9] 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.
[10] David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag., 82(3):163–173, 2009.
[11] David Speyer and Bernd Sturmfels. Tropical mathematics. Math. Mag., 82(3):163–173, 2009.
[12] Yen-Lung Tsai. Working with tropical meromorphic functions of one variable. Taiwanese J. Math., 16(2):691–712, 2012.
[13] Yen-Lung Tsai. Working with tropical meromorphic functions of one variable. Taiwanese J. Math., 16(2):691–712, 2012. |
| Description: | 碩士
國立政治大學
應用數學研究所
100751006
103 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G1007510061 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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| 006101.pdf | | 2400Kb | Adobe PDF2 | 520 | View/Open |
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