Minkowski's theorem

In mathematics, Minkowski's theorem is the statement that any convex set in Rn which is symmetric with respect to the origin and with volume greater than 2n contains a non-zero lattice point. The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called geometry of numbers.

Formulation

Suppose that L is a lattice of determinant d(L) in the n-dimensional real vector space Rn and S is a convex subset of Rn that is symmetric with respect to the origin, meaning that if x is in S then −x is also in S. Minkowski's theorem states that if the volume of S is strictly greater than 2n d(L), then S must contain at least one lattice point other than the origin.[1]

Example

The simplest example of a lattice is the set Zn of all points with integer coefficients; its determinant is 1. For n = 2 the theorem claims that a convex figure in the plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin. The area bound is sharp: if S is the interior of the square with vertices (±1, ±1) then S is symmetric and convex, has area 4, but the only lattice point it contains is the origin. This observation generalizes to every dimension n.

Applications

A corollary of this theorem is the fact that every class in the ideal class group of a number field K contains an integral ideal of norm not exceeding a certain bound, depending on K, called Minkowski's bound.

Notes

1. ^ Since the set S is symmetric, it would then contain at least three lattice points: the origin 0 and a pair of points ±x, where x ∈ L \ 0.

See also

* Pick's theorem

* Dirichlet's unit theorem

References

* Minkowski's constant on PlanetMath

* Stevenhagen, Peter. Number Rings.

Number Theory

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