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    Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/43899

    Title: 透過代數幾何計算環面上平均場方程解的個數;Counting solutions of the mean field equations on tori via algebraic geometry
    Authors: 林耿立;Keng-Li Lin
    Contributors: 數學研究所
    Keywords: 代數幾何;平均場方程;algebraic geometry;mean field equations
    Date: 2010-06-15
    Issue Date: 2010-12-08 14:25:53 (UTC+8)
    Publisher: 國立中央大學
    Abstract: 關於環面上的奇異平均場方程,在單一Dirac奇異點的係數為4π(2k+1)時林長壽教授與王金龍教授曾給出一個猜想:解的個數恰好等於其拓樸度數。 吾等發現如此的非線性方程原來是代數可積的,並且此計數問題可化約至對某一個仿射多項式系統計算。 我们提出兩個都基於代數幾何的方法來證明此一猜想。第一個方法是基於射影的Bézout定理與殘餘相交理論,運用此一方法我们可以證實該猜想到k≦5的情形。射影化系統中多出之無窮遠解的結構牽涉到某種對稱的組合與精緻的多面體結構,這些結搆似乎與其他數學領域有關並值得進一步研究。 第二個方法是對所有k在仿射方程組上的同倫法。這部分仍存在一些證明上的空白,所以吾等僅提及其想法與一些關鍵點。希望這將會在未來給出該猜想一個直接的證明。 It is conjectured by C.-S. Lin and C.-L. Wang. cite{LW2} that the number $N_k$ solutions for the  singular mean field equation on tori (with the  coefficient $4pi(2k + 1)$ of the delta singularity ) should be equal to its topological degree $k+1$  for each $k in mathbb{N}$ cite{LW2}. We  verifies the case of $k = 4$ and $5$ via intersection theory. In these cases, it shows that  the solution toward the general cases involves  some symmetrically combinatorial and delicately  polyhedron structures.
    Appears in Collections:[數學研究所] 博碩士論文

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