Summary
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 γn for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn with unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then √γn is the maximum of λ1(L) over all such lattices L.
The square root in the definition of the Hermite constant is a matter of historical convention.
Alternatively, the Hermite constant γn can be defined as the square of the maximal systole of a flat n-dimensional torus of unit volume.
Example
editThe Hermite constant is known in dimensions 1–8 and 24.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 |
|---|---|---|---|---|---|---|---|---|---|
For n = 2, one has γ2 = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers.[1]
Estimates
editIt is known that[2]
A stronger estimate due to Hans Frederick Blichfeldt[3] is[4]
where is the gamma function.
See also
editReferences
edit- 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.