查看“︁几何数论”︁的源代码

在[[数论]]中，'''几何数论'''（{{lang-en|Geometry of numbers}}）研究[[凸体]]和在n[[维]]空间整数点向量问题。几何数论于1910由[[赫尔曼·闵可夫斯基]]创立。几何数论和数学其它领域有密切的关系，尤其研究在[[泛函分析]]和[[丢番图逼近]]中，对[[有理数]]向[[无理数]]逼近问题。<ref>Schmidt's books. Grötschel et alia, Lovász et alia, Lovász.</ref>

==闵可夫斯基的结果==
{{main|闵可夫斯基定理}}
*闵可夫斯基定理，有时也被称为闵可夫斯基第一定理：
:假设Γ是在n[[维]][[欧氏空间]]R<sup>''n''</sup>的[[格]]和''K''是中心对称[[凸体]]，<math>vol(K)>2^nvol(R^n/\Gamma)</math> ，则''K''包含Γ非零的向量。
*闵可夫斯基第二[[定理]]，是他的第一定理加强。定义''K''数字λ[[最大下界]]，为 λ<sub>''k''</sub>，称为[[连续]]最低。
则λK在Γ中ķ[[线性无关]]，则有：
:<math>\lambda_1\lambda_2\cdots\lambda_n vol(K)\le 2^n vol(R^n/\Gamma).</math>

==近现代几何数论研究==
在1930年至1960年的很多数论学家取得了很多成果（包括[[路易·莫德尔]]，[[哈罗德·达文波特]]和[[卡尔·西格尔|卡尔·路德维希·西格尔]]）。近年来，Lenstra，奥比昂，巴尔维诺克对组合理论的扩展对一些凸体的格数量进行了列举。
*[[施密特]]子空间定理
*在几何数论的子空间定理，由沃尔夫冈·施密特在1972年证明
*设''n''是正整数，如果''n''个''n''维线性型''L''<sub>1</sub>,...,''L''<sub>''n''</sub>都具有代数[[系数]]，並且是[[线性无关]]的，那么对于任何给定的[[实数]]ε> 0，所有满足条件: <math>|L_1(x)\cdots L_n(x)|<|x|^{-\epsilon}</math> 的n维非零整数点x都在有限多个Q<sup>''n''</sup>的真[[子空间]]内。

==对泛函分析的影响==
始于闵可夫斯基的几何数论在[[泛函分析]]上产生深远的影响。闵可夫斯基证明，对称[[凸体]]诱导[[有限维向量空间]]的[[范数]]。闵可夫斯基定理由[[柯尔莫哥洛夫]]推广到[[拓扑向量空间]]。柯尔莫哥洛夫的定理证明[[有界集合|有界]]闭对称凸集生成[[Banach空间]]的[[拓扑]]。当前Kalton et alia. Gardner对[[星形集]]和[[非凸集]]取得了一些成果。

==参考文献==
{{reflist}}

==延伸阅读==
* Matthias Beck, Sinai Robins. ''Computing the continuous discretely: Integer-point enumeration in polyhedra'', Undergraduate texts in mathematics, Springer, 2007.
*{{cite journal|author = Enrico Bombieri|coauthors = Vaaler, J.|title = On Siegel's lemma|journal = Inventiones Mathematicae|volume = 73|issue = 1|date = Feb 1983|pages = 11–32|url = http://www.springerlink.com/content/k55042224131lp42|doi = 10.1007/BF01393823}}{{Dead link|date=2020年2月 |bot=InternetArchiveBot |fix-attempted=yes }}
*{{cite book|author=Enrico Bombieri and Walter Gubler|title=Heights in Diophantine Geometry|publisher=Cambridge U. P.|year=2006}}
* J. W. S. Cassels. ''An Introduction to the Geometry of Numbers''. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).
* John Horton Conway and N. J. A. Sloane, ''Sphere Packings, Lattices and Groups'', Springer-Verlag, NY, 3rd ed., 1998.
*R. J. Gardner, ''Geometric tomography,'' Cambridge University Press, New York, 1995. Second edition: 2006.
*P. M. Gruber, ''Convex and discrete geometry,'' Springer-Verlag, New York, 2007.
*P. M. Gruber, J. M. Wills (editors), ''Handbook of convex geometry. Vol. A. B,'' North-Holland, Amsterdam, 1993.
*M. Grötschel, L. Lovász, A. Schrijver: ''Geometric Algorithms and Combinatorial Optimization'', Springer, 1988
*{{cite book| author = Hancock, Harris| title = Development of the Minkowski Geometry of Numbers| url = https://archive.org/details/developmentofmin0000harr| year = 1939| publisher = Macmillan}} (Republished in 1964 by Dover.)
* Edmund Hlawka, Johannes Schoißengeier, Rudolf Taschner. ''Geometric and Analytic Number Theory''. Universitext. Springer-Verlag, 1991.
* {{citation|last1=Kalton|first1=Nigel J.|last2=Peck|first2=N. Tenney|last3=Roberts|first3=James W.| title = An F-space sampler| series = London Mathematical Society Lecture Note Series, 89| publisher = Cambridge University Press| publication-place = Cambridge| year = 1984| pages = xii+240| isbn = 0-521-27585-7 | mr = 0808777}}
* C. G. Lekkerkererker. ''Geometry of Numbers''. Wolters-Noordhoff, North Holland, Wiley. 1969.
* {{cite journal | author = Lenstra, A. K.; Lenstra, H. W., Jr.; Lovász, L. | title = Factoring polynomials with rational coefficients | url = https://archive.org/details/sim_mathematische-annalen_1982-12_261_4/page/515 | journal = Mathematische Annalen | volume = 261 | year = 1982 | issue = 4 | pages = 515–534 | doi = 10.1007/BF01457454 | mr = 0682664}}
*L. Lovász: ''An Algorithmic Theory of Numbers, Graphs, and Convexity'', CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
*{{Springer|id=G/g044350|title=Geometry of numbers|first=A.V. |last=Malyshev}}
*{{Citation | last1=Minkowski | first1=Hermann | title=Geometrie der Zahlen | url=http://www.archive.org/details/geometriederzahl00minkrich | publisher=R. G. Teubner | location=Leipzig and Berlin | mr=0249269 | year=1910}}
* Wolfgang M. Schmidt. ''Diophantine approximation''. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
* Wolfgang M. Schmidt.''Diophantine approximations and Diophantine equations'', Lecture Notes in Mathematics, Springer Verlag 2000.
*{{cite book| author = Siegel, Carl Ludwig| title = Lectures on the Geometry of Numbers| url = https://archive.org/details/lecturesongeomet0000sieg| year = 1989| publisher = Springer-Verlag}}
* Rolf Schneider, ''Convex bodies: the Brunn-Minkowski theory,'' Cambridge University Press, Cambridge, 1993.
* Anthony C. Thompson, ''Minkowski geometry,'' Cambridge University Press, Cambridge, 1996.
{{refend}}
{{数学领域}}
[[Category:数论]]
[[Category:离散几何]]