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題名 熱帶橢圓曲線之研究
On Tropical Elliptic Curves
作者 黃明怡
Huang, Ming Yi
貢獻者 蔡炎龍
Tsai, Yen lung
黃明怡
Huang, Ming Yi
關鍵詞 熱帶幾何
橢圓曲線
因子理論
Picard 群
tropical geometry
elliptic curve
divisor theory
Picard group
日期 2014
上傳時間 3-Feb-2015 10:24:54 (UTC+8)
摘要 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.
參考文獻 [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.
描述 碩士
國立政治大學
應用數學研究所
100751006
103
資料來源 http://thesis.lib.nccu.edu.tw/record/#G1007510061
資料類型 thesis
dc.contributor.advisor 蔡炎龍zh_TW
dc.contributor.advisor Tsai, Yen lungen_US
dc.contributor.author (Authors) 黃明怡zh_TW
dc.contributor.author (Authors) Huang, Ming Yien_US
dc.creator (作者) 黃明怡zh_TW
dc.creator (作者) Huang, Ming Yien_US
dc.date (日期) 2014en_US
dc.date.accessioned 3-Feb-2015 10:24:54 (UTC+8)-
dc.date.available 3-Feb-2015 10:24:54 (UTC+8)-
dc.date.issued (上傳時間) 3-Feb-2015 10:24:54 (UTC+8)-
dc.identifier (Other Identifiers) G1007510061en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/73287-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 100751006zh_TW
dc.description (描述) 103zh_TW
dc.description.abstract (摘要) 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。zh_TW
dc.description.abstract (摘要) 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.en_US
dc.description.tableofcontents 口試委員會審定書 i
中文摘要 ii
Abstract iii
Contents iv
List of Figures vi
1 Introduction 1
2 Tropical Geometry 2
2.1 TropicalSemifield................................. 2
2.2 TropicalPolynomial ............................... 3
2.3 TropicalCurve .................................. 6

3 Divisor Theory on Tropical Geometry 16
3.1 TropicalDivisors ................................. 16
3.2 SpecialDivisors.................................. 18
3.3 TropicalPicardGroup .............................. 20

4 Tropical Elliptic Curves 22
4.1 ClassicalEllipticCurves ............................. 22
4.2 TropicalEllipticCurves.............................. 24
4.3 Grouplaw..................................... 27

5 Classical Elliptic Curves and the Cryptogrophy 33
5.1 EllipticCurveCryptogrophy ........................... 33
5.2 TropicalEllipticCurveCryptogrophy ...................... 34
5.3 SecurityIssue................................... 35

6 Conclusion 36
Bibliography 37		zh_TW
dc.format.extent 2458433 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G1007510061en_US
dc.subject (關鍵詞) 熱帶幾何zh_TW
dc.subject (關鍵詞) 橢圓曲線zh_TW
dc.subject (關鍵詞) 因子理論zh_TW
dc.subject (關鍵詞) Picard 群zh_TW
dc.subject (關鍵詞) tropical geometryen_US
dc.subject (關鍵詞) elliptic curveen_US
dc.subject (關鍵詞) divisor theoryen_US
dc.subject (關鍵詞) Picard groupen_US
dc.title (題名) 熱帶橢圓曲線之研究zh_TW
dc.title (題名) On Tropical Elliptic Curvesen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [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.		zh_TW