希尔伯特第二十一问题

希尔伯特第二十一问题是希尔伯特的23个问题之一：给定 $P_{1},\ldots ,P_{n}\in \mathbb {P} ^{1}(\mathbb {C} ),\Omega :=\mathbb {C} -\{P_{1},\ldots ,P_{n}\}$ 及一个线性表示 $\rho :\pi _{1}(\Omega ,x_{0})\rightarrow GL(m,\mathbb {C} )$ (给定 $x_{0}\in \Omega$ ），是否存在一组 $\Omega$ 上的Fuchs方程，使得其单值群由 $\rho$ 给出？

现况

此问题的答案决定于其表述：如果我们容许明显的奇异点（即：其单值群是平凡的），并在复流形上的向量丛及其联络的意义下理解Fuchs方程，则答案是肯定的；否则存在反例。这是L. Plemelj、G. Birkhoff、I. Lappo-Danilevskij、P. Deligne与A. Bolibrukh等数学家的工作。^[1]^[2]^[3]^[4]^[5]

此问题有时亦称为黎曼-希尔伯特问题。数学家柏原正树与Zoghman Mebkhout已借助D-模的抽象语言将此结果推广到高维情形，称作黎曼-希尔伯特对应。

文献

A. Beauville, Equations différentielles à points singuliers réguliers d'apres Bolybrukh, Sem. Bourbaki , 1992/3（1993） pp. 103–120
A. Borel Algebraic D-modules ISBN 0-12-117740-8
P. Deligne, Equations differentials a points singuliers reguliers, Springer Lecture notes in mathematics 163 (1970).
M. Kashiwara, Faiseaux constructibles et systems holonomes d'equations aux derivees partielles lineaires a points singuliers reguliers, Se. Goulaouic-Schwartz, 1979-80, Exp. 19.
Z. Mebkhout, Sur le probleme de Hilbert-Riemann, Lecture notes in physics 129 (1980) 99-110.

外部链接

M. Hazewinkel, Hilbert problems, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4

^ Anosov, D. V.; Bolibruch, A. A. The Riemann-Hilbert Problem. Aspects of Mathematics 22. Wiesbaden: Vieweg+Teubner Verlag http://link.springer.com/10.1007/978-3-322-92909-9. 1994. ISBN 9783322929112. doi:10.1007/978-3-322-92909-9. 缺少或|title=为空 (帮助)
^ Bolibrukh, A A. The Riemann-Hilbert problem. Russian Mathematical Surveys. 1990-04-30, 45 (2): 1–58. ISSN 0036-0279. doi:10.1070/RM1990v045n02ABEH002350.
^ Bandemer, H. J. Plemelj, Problems in the Sense of Riemann and Klein (Interscience Tracts in Pure and Applied Mathematics, Nr. 16). 173 S. m. 15 Fig. New York/London/Sydney 1964. John Wiley & Sons Inc. Preis geb. 60 s. net. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik. 1965, 45 (1): 67–67. ISSN 0044-2267. doi:10.1002/zamm.19650450117.
^ Bolibrukh, A. A. On sufficient conditions for the positive solvability of the Riemann-Hilbert problem. Mathematical Notes. 1992-2, 51 (2): 110–117. ISSN 0001-4346. doi:10.1007/BF02102113 （英语）. 请检查|date=中的日期值 (帮助)
^ Kostov, Vladimir Petrov. On the Deligne-Simpson problem. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 1999-10, 329 (8): 657–662. ISSN 0764-4442. doi:10.1016/s0764-4442(00)88212-9.

[1] Anosov, D. V.; Bolibruch, A. A. The Riemann-Hilbert Problem. Aspects of Mathematics 22. Wiesbaden: Vieweg+Teubner Verlag http://link.springer.com/10.1007/978-3-322-92909-9. 1994. ISBN 9783322929112. doi:10.1007/978-3-322-92909-9. 缺少或|title=为空 (帮助)

[2] Bolibrukh, A A. The Riemann-Hilbert problem. Russian Mathematical Surveys. 1990-04-30, 45 (2): 1–58. ISSN 0036-0279. doi:10.1070/RM1990v045n02ABEH002350.

[3] Bandemer, H. J. Plemelj, Problems in the Sense of Riemann and Klein (Interscience Tracts in Pure and Applied Mathematics, Nr. 16). 173 S. m. 15 Fig. New York/London/Sydney 1964. John Wiley & Sons Inc. Preis geb. 60 s. net. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik. 1965, 45 (1): 67–67. ISSN 0044-2267. doi:10.1002/zamm.19650450117.

[4] Bolibrukh, A. A. On sufficient conditions for the positive solvability of the Riemann-Hilbert problem. Mathematical Notes. 1992-2, 51 (2): 110–117. ISSN 0001-4346. doi:10.1007/BF02102113 （英语）. 请检查|date=中的日期值 (帮助)

[5] Kostov, Vladimir Petrov. On the Deligne-Simpson problem. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics. 1999-10, 329 (8): 657–662. ISSN 0764-4442. doi:10.1016/s0764-4442(00)88212-9.

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