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    題名: 最大係數熱帶多項式及其應用
    Largest-coefficient Tropical Polynomials and Their Applications
    作者: 林如苹
    貢獻者: 蔡炎龍
    林如苹
    關鍵詞: 熱帶幾何
    熱帶多項式
    最大係數熱帶多項式
    熱帶代數基本定理
    日期: 2008
    上傳時間: 2009-09-19 12:08:25 (UTC+8)
    摘要: 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.
    參考文獻: [1] Lucia Caporaso and Joe Harris. Counting plane curves of any genus. Inventiones Mathematicae, 131(2):345{392, 2 1998.
    [2] Andreas Gathmann. Tropical algebraic geometry. Jahresber. Deutsch. Math.-Verein., 108(1):3{32, 2006.
    [3] Nathan Grigg and Nathan Manwaring. An elementary proof
    of the fundamental theorem of tropical algebra. Preprint at
    arXiv:math.CO/0701.2591, February 2008.
    [4] Nathan B. Grigg. Factorization of Tropical Polynomials in One and Several Variables. Honor`s thesis, Brigham Young University, June 2007.
    [5] Grigory Mikhalkin. Counting curves via lattice paths in polygons. C.R. Math. Acad. Sci. Paris, 336(8):629{634, 2003.
    [6] Grigory Mikhalkin. Enumerative tropical algebraic geometry in R2. J.Amer. Math. Soc., 18(2):313{377 (electronic), 2005.
    [7] Grigory Mikhalkin. Tropical geometry and its applications. In Interna-tional Congress of Mathematicians. Vol. II, pages 827{852. Eur. Math.
    Soc., Zurich, 2006.
    [8] Jurgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. Contemporary Mathematics, 377:289{317,2005.
    [9] David Speyer and Bernd Sturmfels. The tropical grassmannian. Ad-vances in Geometry, 4:389{411, 2004.
    [10] David Speyer and Bernd Sturmfels. Tropical mathematics. Preprint at arXiv:math.CO/0408099, 2004.
    描述: 碩士
    國立政治大學
    應用數學研究所
    95972002
    97
    資料來源: http://thesis.lib.nccu.edu.tw/record/#G0095972002
    資料類型: thesis
    顯示於類別:[應用數學系] 學位論文

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