Hermite constant

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In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.

Contents

The constant for integers is defined as follows. For a lattice in Euclidean space with unit covolume, i.e. , let denote the least length of a nonzero element of . Then is the maximum of over all such lattices .

The square root in the definition of the Hermite constant is a matter of historical convention.

Alternatively, the Hermite constant can be defined as the square of the maximal systole of a flat -dimensional torus of unit volume.

Example

A hexagonal lattice with unit covolume (the area of the quadrilateral is 1). Both arrows are minimum non-zero elements for
n
-
1
{\displaystyle n-1}
with length
l
n
=
g
n
=
2
/
3
{\textstyle \lambda _{n}={\sqrt {\gamma _{n}}}={\sqrt {2/{\sqrt {3}}}}}
. Hexagonal lattice.png
A hexagonal lattice with unit covolume (the area of the quadrilateral is 1). Both arrows are minimum non-zero elements for with length .

The Hermite constant is known in dimensions 1–8 and 24.

n1234567824

For , one has . This value is attained by the hexagonal lattice of the Eisenstein integers, scaled to have a fundamental parallelogram with unit area. [1]

The constants for the missing values are conjectured. [2]

Estimates

It is known that [3]

A stronger estimate due to Hans Frederick Blichfeldt [4] is [5]

where is the gamma function.

See also

References

  1. Cassels (1971) p. 36
  2. Leon Mächler; David Naccache (2022). "A Conjecture on Hermite Constants". Cryptology ePrint Archive.
  3. Kitaoka (1993) p. 36
  4. Blichfeldt, H. F. (1929). "The minimum value of quadratic forms, and the closest packing of spheres". Math. Ann. 101: 605–608. doi:10.1007/bf01454863. JFM   55.0721.01. S2CID   123648492.
  5. Kitaoka (1993) p. 42