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論文名稱 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 viaalgebraic geometry)
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摘要(中) 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。
摘要(英) 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.
關鍵字(中) ★ 代數幾何
★ 平均場方程
關鍵字(英) ★ algebraic geometry
★ mean field equations
論文目次 1 Introduction 1
2 Reduction from PDE to polynomial systems 3
2.1 Fundamental results . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 The canonical representation of solutions . . . . . . . . . . . 6
2.3 The correspondence theorem . . . . . . . . . . . . . . . . . . 7
2.4 The case k = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Approach by intersection theory 17
3.1 The residual intersection formula . . . . . . . . . . . . . . . . 17
3.2 The case k = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 The case k = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Some partial results for the general case . . . . . . . . . . . . 34
3.5 The symmetry and combinatorics behind the system . . . . . 38
4 An approach by the homotopy method 41
References 42
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 in two dimensions, Comm. Partial Differential
Equations 16 (1991), 1223-1254.
D. Bartolucci and G. Tarantello; The Liouville
Equation with Singular Data: A Concentration
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1(2002), pp. 161-180.
---{}---; Topological degree for a Liouville type
equation with a singular source, preprint.
C.-C.~Chen and C.-S.~Lin and Guofang Wang;
Concentration Phenomena of Two-Vortex Solutions in a
Chern-Simons Model, Ann. Scuola Norm. Sup. Pisa Cl.
Sci.(5), Vol. III (2004), pp. 367-397.
K. S. Chou and Tom Yau-Heng Wan;
Asymptotic radial symmetry for solutions of 
Delta u+e^u=0 in a punctured disc, Pacific J.
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B.H. Dayton and Z. Zeng;
Computing the Multiplicity Structure in Solving
Polynomial Systems, Proceedings of the 2005
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 Gren.;3. Folge, Bd 2, Springer-Verlag 1984.
Y.Y. Li and I. Shafrir, Blow up analysis for
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Ind. Univ. Math. J. 43 (1994) (4), pp. 1255–1270.
Tien-Yien Li, Tim Sauer and James A.Yorke; Numberical
solution of a class of deficient polynomial
 systems, SIAM J Number. Anal., Vol24, No. 2,
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C.-S.~Lin and C.-L.~Wang; Elliptic functions, Green
 functions and the mean field equations on tori,
 to appear in Annals of Mathematics, accepted in
 April 2008. arXiv:math/0608358.
---{}---; A function theoretic view of the Mean
 field equations on tori, Proceeding of the
 International Conference on Geometric Analysis
 (TIMS, Taipei 2007), International Press 2008.
---{}---; Singular mean field equations on tori,
 preprint 2009.
http://www.valdostamuseum.org/hamsmith/Weyl.html.
指導教授 王金龍(Chin-LungWang) 審核日期 2010-6-15
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