“Minkowski's second theorem”的意思、由来-开放百科全书

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Minkowski's second theorem

释义

Setting
Statement of the theorem
Proof of the theorem
References)

In mathematics, Minkowski's second theorem is a result in the Geometry of numbers about the values taken by a norm on a lattices and the volume of its fundamental cell.

Setting

Let {{math|K}} be a closed convex centrally symmetric body of positive finite volume in {{math|n}}-dimensional Euclidean space {{math|ℝⁿ}}. The gauge^[1] or distance^[2]^[3] Minkowski functional {{math|g}} attached to {{math|K}} is defined by

Conversely, given a norm {{math|g}} on {{math|ℝⁿ}} we define {{math|K}} to be

Let {{math|Γ}} be a lattice in {{math|ℝⁿ}}. The successive minima of {{math|K}} or {{math|g}} on {{math|Γ}} are defined by setting the {{math|k}}th successive minimum {{math|λ_k}} to be the infimum of the numbers {{math|λ}} such that {{math|λK}} contains {{math|k}} linearly-independent vectors of {{math|Γ}}. We have {{math|0 < λ₁ ≤ λ₂ ≤ ... ≤ λ_n < ∞}}.

Statement of the theorem

The successive minima satisfy^[4]^[5]^[6]

Proof of the theorem

A basis of linearly independent lattice vectors {{math| b₁ , b₂ , ... b_n }} can be defined by {{math|1=g(b_j) = λ_j}} .

The lower bound is proved by considering the convex polytope {{math|2n}} with vertices at {{math|±b_j/ λ_j }}, which has an interior enclosed by {{math|K}} and a volume which is {{math|2ⁿ/n!λ₁ λ₂...λ_n}} times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by {{math|λ_j }} along each basis vector to obtain {{math|2ⁿ}} {{math|n}}-simplices with lattice point vectors).

To prove the upper bound, consider functions {{math|f_j(x) }}sending points {{math|x}} in to the centroid of the subset of points in that can be written as for some real numbers . Then the coordinate transform has a Jacobian determinant . If and are in the interior of and (with ) then with , where the inclusion in (specifically the interior of ) is due to convexity and symmetry. But lattice points in the interior of are, by definition of , always expressible as a linear combination of , so any two distinct points of cannot be separated by a lattice vector. Therefore, must be enclosed in a primitive cell of the lattice (which has volume ) , and consequently .

References)

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

{{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 S. Chandrasekharan | zbl=0691.10021 }}

2 : Geometry of numbers|Hermann Minkowski

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