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).