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 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
The Hermite constant is known in dimensions 1–8 and 24.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 24 |
|---|---|---|---|---|---|---|---|---|---|
| [math]\displaystyle{ \gamma_n^n }[/math] | [math]\displaystyle{ 1 }[/math] | [math]\displaystyle{ \frac 4 3 }[/math] | [math]\displaystyle{ 2 }[/math] | [math]\displaystyle{ 4 }[/math] | [math]\displaystyle{ 8 }[/math] | [math]\displaystyle{ \frac {64} 3 }[/math] | [math]\displaystyle{ 64 }[/math] | [math]\displaystyle{ 2^8 }[/math] | [math]\displaystyle{ 4^{24} }[/math] |
For n = 2, one has γ2 = 2/√3. This value is attained by the hexagonal lattice of the Eisenstein integers.[1]
Estimates
It is known that[2]
- [math]\displaystyle{ \gamma_n \le \left( \frac 4 3 \right)^\frac{n-1}{2}. }[/math]
A stronger estimate due to Hans Frederick Blichfeldt[3] is[4]
- [math]\displaystyle{ \gamma_n \le \left( \frac 2 \pi \right)\Gamma\left(2 + \frac n 2\right)^\frac{2}{n}, }[/math]
where [math]\displaystyle{ \Gamma(x) }[/math] is the gamma function.
See also
References
- 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. https://archive.org/details/springer_10.1007-978-3-642-62035-5.
- Kitaoka, Yoshiyuki (1993). Arithmetic of quadratic forms. Cambridge Tracts in Mathematics. 106. Cambridge University Press. ISBN 0-521-40475-4.
- Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. 1467 (2nd ed.). Springer-Verlag. p. 9. ISBN 3-540-54058-X.
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