查看“︁希爾伯特第二十一問題”︁的源代码

'''希爾伯特第二十一問題'''是[[希爾伯特的23個問題]]之一：給定<math>P_1, \ldots, P_n \in \mathbb{P}^1(\mathbb{C}), \Omega := \mathbb{C} - \{P_1, \ldots, P_n\}</math>及一個[[表示 (群)|線性表示]]<math>\rho : \pi_1(\Omega, x_0) \rightarrow GL(m, \mathbb{C}) </math>(給定<math>x_0 \in \Omega</math>），是否存在一組<math>\Omega </math>上的[[Fuchs方程]]，使得其[[單值群]]由<math>\rho</math>給出？

==現況==
此問題的答案決定於其表述：如果我們容許明顯的奇異點（即：其單值群是平凡的），並在[[複流形]]上的[[向量叢]]及其[[聯絡]]的意義下理解[[Fuchs方程]]，則答案是肯定的；否則存在反例。這是L. Plemelj、G. Birkhoff、I. Lappo-Danilevskij、P. Deligne與A. Bolibrukh等數學家的工作。<ref>{{Cite book|chapter=The Riemann-Hilbert Problem|series=Aspects of Mathematics|url=http://link.springer.com/10.1007/978-3-322-92909-9|publisher=Vieweg+Teubner Verlag|date=1994|location=Wiesbaden|isbn=9783322929112|volume=22|doi=10.1007/978-3-322-92909-9|first=D. V.|last=Anosov|first2=A. A.|last2=Bolibruch}}</ref><ref>{{Cite journal|title=The Riemann-Hilbert problem|url=http://stacks.iop.org/0036-0279/45/i=2/a=R01?key=crossref.47f8f34719b47214e4e37332092b46c3|last=Bolibrukh|first=A A|date=1990-04-30|journal=Russian Mathematical Surveys|issue=2|doi=10.1070/RM1990v045n02ABEH002350|volume=45|pages=1–58|issn=0036-0279}}</ref><ref>{{Cite journal|title=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|url=http://dx.doi.org/10.1002/zamm.19650450117|last=Bandemer|first=H.|date=1965|journal=ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik|issue=1|doi=10.1002/zamm.19650450117|volume=45|pages=67–67|issn=0044-2267}}</ref><ref>{{Cite journal|title=On sufficient conditions for the positive solvability of the Riemann-Hilbert problem|url=http://link.springer.com/10.1007/BF02102113|last=Bolibrukh|first=A. A.|date=1992-2|journal=Mathematical Notes|issue=2|doi=10.1007/BF02102113|volume=51|pages=110–117|language=en|issn=0001-4346}}</ref><ref>{{Cite journal|title=On the Deligne-Simpson problem|url=http://dx.doi.org/10.1016/s0764-4442(00)88212-9|last=Kostov|first=Vladimir Petrov|date=1999-10|journal=Comptes Rendus de l'Académie des Sciences - Series I - Mathematics|issue=8|doi=10.1016/s0764-4442(00)88212-9|volume=329|pages=657–662|issn=0764-4442}}</ref>

此問題有時亦稱為[[黎曼-希爾伯特問題]]。數學家柏原正樹與Zoghman Mebkhout已藉助[[D-模]]的抽象語言將此結果推廣到高維情形，稱作[[黎曼-希爾伯特對應]]。

==文獻==
{{refbegin}}
* 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.
{{refend}}

==外部連結==
* {{springer|author=M. Hazewinkel|id=H/h120080|title=Hilbert problems}}
{{Hilbert's problems}}
[[Category:微分方程]]
[[Category:希尔伯特问题]]