6 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS
Figure 4. Two bases of the hexagonal lattice obtained from the
PQFs in Figure 1.
The volume is called the determinant of the lattice L =
AZd
and denoted by det L.
This is justified, since it does not depend on the choice of the particular basis. For
the PQF Q =
AtA
we have det L =

det Q.
1.2.2. Hermite’s constant. The lattice interpretation of Gauss opened the
door for the use of geometric arguments in the theory of quadratic forms. A classical
example is the following theorem and its proof.
Theorem 1.1 (Hermite, [135]). Every lattice L has a base A = (a1, . . . , ad)
GLd(R), such that
a1···ad
4
3
d(d−1)/4
det L.
Using the dictionary, the theorem gives an upper bound for the arithmetical
minimum, depending on the determinant of a given PQF Q:
Corollary 1.2. λ(Q) (det
Q)1/d
·
(
4
3
)(d−1)/2
for Q
Sd
0
.
The corollary in particular implies the existence of Hermite’s constant
(1.12) Hd = sup
Q∈Sd
0
λ(Q)
(det Q)1/d
.
Hermite’s constant and generalizations have been extensively studied, e.g. in the
context of algebraic number theory and differential geometry. We refer to [32],
[220], [70] and [265] for further reading.
1.2.3. Lattice sphere packings. Hermite’s theorem 1.1 and its proof (cf.
[174]) show how helpful the geometric viewpoint can be to obtain arithmetic results
as Corollary 1.2. The following sphere packing interpretation due to Minkowski
exemplifies this even more.
For a PQF Q =
AtA
and a lattice L =
AZd,
λ(L) =
λ(Q)
2
is called the packing radius of L. Thinking of solid spheres of radius λ(L) around
each lattice point, the spheres do not overlap and form a (lattice) sphere packing
(1.13) L +
λ(L)Bd
= {v + λ(L)x : v L, x
Bd}.
See Figure 5 for portions of two famous examples. Due to this interpretation of the
arithmetical minimum, the number | Min Q| of its representatives is also known as
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