1.1. POSITIVE DEFINITE QUADRATIC FORMS 3
Figure 1. Ellipses given by
x2
+ xy +
y2
1 and
3x2
+ 3xy +
y2
1 .
For PQFs there exists a smallest real number n for which question (1.4) has
integral solutions. This number is called the arithmetical minimum and denoted by
λ(Q) = min
x∈Zd\{0}
Q[x].
Sometimes λ(Q) is referred to as the homogeneous minimum. We have λ(αQ) =
αλ(Q) for a non-negative real scalar α.
A PQF Q defines a real valued strictly convex function on Rd and for λ 0
(1.6) E(Q, λ) = {x
Rd
: Q[x] λ}
is a non-empty ellipsoid with center 0, providing a geometric interpretation of a
PQF. Interpreted geometrically, the arithmetical minimum is the smallest number
λ for which the ellipsoid E(Q, λ) contains an integral point aside of 0. The integral
points x attaining the arithmetical minimum lie on the boundary of the ellipsoid
E(Q, λ(Q)).
Min Q = {x
Zd
: Q[x] = λ(Q)}
is the set of representatives of the arithmetical minimum.
Arithmetical equivalent PQFs Q and Q , as the two forms in (1.5) for example,
have the same arithmetical minimum, and if Q = U
tQU
for U GLd(Z), then the
sets of representatives satisfy Min Q = U Min Q . See Figure 1 for the two ellipses
given by the arithmetical equivalent forms in (1.5).
1.1.4. The space of positive definite quadratic forms. By the Sylvester
criterion, Q Sd is positive definite if and only if the upper left minors Qk =
(qij )i,j=1,...,k of Q have positive determinants. Thus
Sd
0
= {Q
Sd
: det Qk 0, k = 1, . . . , d}
is given by d polynomial inequalities. It is not hard to see that Sd
0
is an open (full
dimensional) convex cone in
Sd
with apex 0. In particular for Q
Sd
0
, the open
ray {λQ : λ 0} is contained in Sd
0
as well. The cone Sd
0
lies in the halfspace
{Q
Sd
: Q, idd = trace Q 0}, where idd denotes the identity (matrix). The
set of PQFs
Sd
0
can be thought of as a union of sections {Q
Sd
0
: trace Q = C}
of constant trace, which are dilates of each other. For example, the intersection of
S2
0
(identified with
R3)
with a plane of constant trace {Q
S2
: trace Q = C} is
a planar disc (ellipse) for C 0 (see Figure 2).
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