Hermite constant

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

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 with length .

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

n 1 2 3 4 5 6 7 8 24

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
  • Cassels, J.W.S. (1997). An Introduction to the Geometry of Numbers. Classics in Mathematics (Reprint of 1971 ed.). Springer-Verlag. ISBN 978-3-540-61788-4.
  • Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. Vol. 106. Cambridge University Press. ISBN 0-521-40475-4. Zbl 0785.11021.
  • Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed.). Springer-Verlag. p. 9. ISBN 3-540-54058-X. Zbl 0754.11020.