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In mathematics, '''Minkowski's second theorem''' is a result in the [[Geometry of numbers]] about the values taken by a quadratic form on a lattice and the volume of its fundamental cell. | |||
==Setting== | |||
Let ''K'' be a [[closed set|closed]] [[convex set|convex]] [[central symmetry|centrally symmetric]] body of positive finite volume in ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>. The ''gauge''<ref>Siegel (1989) p.6</ref> or ''distance''<ref>Cassels (1957) p.154</ref><ref>Cassels (1971) p.103</ref> [[Minkowski functional]] ''g'' attached to ''K'' is defined by | |||
:<math>g(x) = \inf\{\lambda \in \mathbb{R} : x \in \lambda K \} . </math> | |||
Conversely, given a quadratic form ''q'' on '''R'''<sup>''n''</sup> we define ''K'' to be | |||
:<math>K = \{ x \in \mathbb{R}^n : q(x) \le 1 \} . </math> | |||
Let Γ be a [[Lattice (group)|lattice]] in '''R'''<sup>''n''</sup>. The '''successive minima''' of ''K'', ''g'' or ''q'' on Γ are defined by setting the ''k''-th successive minimum λ<sub>''k''</sub> to be the [[infimum]] of the numbers λ such that λ''K'' contains ''k'' linearly independent vectors of Γ. We have 0 < λ<sub>1</sub> ≤ λ<sub>2</sub> ≤ ... ≤ λ<sub>''n''</sub> < ∞. | |||
==Statement of the theorem== | |||
The successive minima satisfy<ref>Cassels (1957) p.156</ref><ref>Cassels (1971) p.203</ref><ref>Siegel (1989) p.57</ref> | |||
:<math>\frac{2^n}{n!} \mathrm{vol}(\mathbb{R}^n/\Gamma) \le \lambda_1\lambda_2\cdots\lambda_n \mathrm{vol}(K)\le 2^n \mathrm{vol}(\mathbb{R}^n/\Gamma).</math> | |||
==References== | |||
{{reflist}} | |||
* {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=An introduction to Diophantine approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=[[Cambridge University Press]] | year=1957 | zbl=0077.04801 }} | |||
* {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=An Introduction to the Geometry of Numbers | series=Classics in Mathematics | publisher=[[Springer-Verlag]] | edition=Reprint of 1971 | year=1997 | isbn=978-3-540-61788-4 }} | |||
* {{cite book | first=Melvyn B. | last=Nathanson | title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets | volume=165 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | year=1996 | isbn=0-387-94655-1 | zbl=0859.11003 | pages=180–185 }} | |||
* {{cite book | last=Schmidt | first=Wolfgang M. | authorlink=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=[[Springer-Verlag]] | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020 | page=6 }} | |||
* {{cite book | first=Carl Ludwig | last=Siegel | authorlink=Carl Ludwig Siegel | title=Lectures on the Geometry of Numbers | publisher=[[Springer-Verlag]] | year=1989 | isbn=3-540-50629-2 | editor=Komaravolu Chandrasekharan | zbl=0691.10021 }} | |||
[[Category:Geometry of numbers| ]] | |||
{{numtheory-stub}} | |||
Revision as of 11:10, 24 July 2013
In mathematics, Minkowski's second theorem is a result in the Geometry of numbers about the values taken by a quadratic form on a lattice and the volume of its fundamental cell.
Setting
Let K be a closed convex centrally symmetric body of positive finite volume in n-dimensional Euclidean space Rn. The gauge[1] or distance[2][3] Minkowski functional g attached to K is defined by
Conversely, given a quadratic form q on Rn we define K to be
Let Γ be a lattice in Rn. The successive minima of K, g or q on Γ are defined by setting the k-th successive minimum λk to be the infimum of the numbers λ such that λK contains k linearly independent vectors of Γ. We have 0 < λ1 ≤ λ2 ≤ ... ≤ λn < ∞.
Statement of the theorem
The successive minima satisfy[4][5][6]
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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