几何数论
在数论中,几何数论(英语: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。[1]
闵可夫斯基的结果
- 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Rn的格和K是中心对称凸体, ,则K包含Γ非零的向量。
则λK在Γ中ķ线性无关,则有:
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·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。
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
- ^ Schmidt's books. Grötschel et alia, Lovász et alia, Lovász.
延伸阅读
- Matthias Beck, Sinai Robins. Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate texts in mathematics, Springer, 2007.
- Enrico Bombieri; Vaaler, J. On Siegel's lemma. Inventiones Mathematicae. Feb 1983, 73 (1): 11–32. doi:10.1007/BF01393823. [永久失效链接]
- Enrico Bombieri and Walter Gubler. Heights in Diophantine Geometry. Cambridge U. P. 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
- Hancock, Harris. Development of the Minkowski Geometry of Numbers. Macmillan. 1939. (Republished in 1964 by Dover.)
- Edmund Hlawka, Johannes Schoißengeier, Rudolf Taschner. Geometric and Analytic Number Theory. Universitext. Springer-Verlag, 1991.
- Kalton, Nigel J.; Peck, N. Tenney; Roberts, James W., An F-space sampler, London Mathematical Society Lecture Note Series, 89, Cambridge: Cambridge University Press: xii+240, 1984, ISBN 0-521-27585-7, MR 0808777
- C. G. Lekkerkererker. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.
- Lenstra, A. K.; Lenstra, H. W., Jr.; Lovász, L. Factoring polynomials with rational coefficients. Mathematische Annalen. 1982, 261 (4): 515–534. MR 0682664. doi:10.1007/BF01457454.
- L. Lovász: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
- Malyshev, A.V., Geometry of numbers, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
- Minkowski, Hermann, Geometrie der Zahlen, Leipzig and Berlin: R. G. Teubner, 1910, MR 0249269
- 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.
- Siegel, Carl Ludwig. Lectures on the Geometry of Numbers. Springer-Verlag. 1989.
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
- Anthony C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.