2 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS
For example, the quadratic forms
(1.1)
x2
+
y2
and
x2
+ xy +
y2
are identified with
(1.2)
1 0
0 1
and
1
1
2
1
2
1
.
The space Sd is a
(
d+1
2
)
-dimensional real vector space. It becomes a Euclidean
space with the usual inner product
(1.3) Q, Q =
d
i,j=1
qij qij = trace(Q · Q ).
So in particular, we may identify
Sd
with
R(d+1).2
With the introduced terminology, many classical problems involving qua-
dratic forms fit into the following framework: Given a quadratic form Q Sd and
n N,
(1.4) does Q[x] = n have integral solutions x
Zd
?
1.1.2. Arithmetical equivalence. When dealing with this problem in the
binary case (d = 2), Lagrange noticed that he could study the question up to
invertible, linear integral substitutions. Two quadratic forms Q, Q
Sd
are called
arithmetically (or integral) equivalent, if there exists a U in the group
GLd(Z) = {U
Zd×d
: | det U| = 1}
such that
Q = U
tQU.
Arithmetically equivalent forms provide the same set of values on integral vectors,
that is, Q [Zd] = Q[Zd], because Q [x] = Q[U x] and U Zd = Zd.
For example, the form
3x2
+ 3xy +
y2
is arithmetically equivalent to
x2
+ xy +
y2
because
(1.5)
−1 2
0 1
1
1
2
1
2
1
−1 0
2 1
=
3
3
2
3
2
1
.
Since for problem (1.4) it is sufficient to deal with one representative of each
equivalence class, it is desirable to know a unique form, or at least a small subset
of each class. These questions are addressed with in the theory of reduction (see
Chapter 2).
1.1.3. Positive definite quadratic forms. The theory of binary quadratic
forms, first systematically studied by Lagrange [162], splits into two branches,
depending on whether the forms are indefinite or definite. Because the roots of
the lattice sphere packing problem are to be found in the latter case, we restrict
ourselves to the definite case from now on.
A quadratic form, respectively real symmetric matrix Q
Sd,
is positive defi-
nite, if Q[x] 0 for all x
Rd
\{0}. The set of all positive definite quadratic forms
(PQFs from now on), respectively matrices, is denoted by
Sd
0
. Negative definite
forms are just the negative of positive forms and most of what follows just differs
by sign.
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