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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 12:10, 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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Template:Numtheory-stub

  1. ↑ Siegel (1989) p.6
  2. ↑ Cassels (1957) p.154
  3. ↑ Cassels (1971) p.103
  4. ↑ Cassels (1957) p.156
  5. ↑ Cassels (1971) p.203
  6. ↑ Siegel (1989) p.57
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