# 几何数论

## 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：

• 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 λk，称为连续最低。

${\displaystyle \lambda _{1}\lambda _{2}\cdots \lambda _{n}vol(K)\leq 2^{n}vol(R^{n}/\Gamma ).}$

## 近现代几何数论研究

• 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
• n是正整数，如果nn维线性型L1,...,Ln都具有代数系数，并且是线性无关的，那么对于任何给定的实数ε> 0，所有满足条件: ${\displaystyle |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\epsilon ))$ 的n维非零整数点x都在有限多个Qn的真子空间内。

## 参考文献

1. ^ Schmidt's books. Grötschel et alia, Lovász et alia, Lovász.
• 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, (编) Hazewinkel, Michiel, 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.