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 |
|---|---|---|---|---|---|---|---|---|---|
| γ n n {\displaystyle \gamma _{n}^{n}} | 1 {\displaystyle 1} | 4 3 {\displaystyle {\frac {4}{3}}} | 2 {\displaystyle 2} | 4 {\displaystyle 4} | 8 {\displaystyle 8} | 64 3 {\displaystyle {\frac {64}{3}}} | 64 {\displaystyle 64} | 2 8 {\displaystyle 2^{8}} | 4 24 {\displaystyle 4^{24}} |
For n = 2, one has γ2 = 2/√3. 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 n values are conjectured.2
Estimates
It is known that3
γ n ≤ ( 4 3 ) n − 1 2 . {\displaystyle \gamma _{n}\leq \left({\frac {4}{3}}\right)^{\frac {n-1}{2}}.}A stronger estimate due to Hans Frederick Blichfeldt4 is5
γ n ≤ ( 2 π ) Γ ( 2 + n 2 ) 2 n , {\displaystyle \gamma _{n}\leq \left({\frac {2}{\pi }}\right)\Gamma \left(2+{\frac {n}{2}}\right)^{\frac {2}{n}},}where Γ ( x ) {\displaystyle \Gamma (x)} is the gamma function.
See also
- 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.
References
Cassels (1971) p. 36 ↩
Leon Mächler; David Naccache (2022). "A Conjecture on Hermite Constants". Cryptology ePrint Archive. https://eprint.iacr.org/2022/677 ↩
Kitaoka (1993) p. 36 ↩
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. /wiki/Hans_Frederick_Blichfeldt ↩
Kitaoka (1993) p. 42 ↩