The Hilbert basis of a convex cone C is a minimal set of integer vectors such that every integer vector in C is a conical combination of the vectors in the Hilbert basis with integer coefficients.
Given a lattice
L
⊂
Z
d
and a convex polyhedral cone with generators
a
1
,
…
,
a
n
∈
Z
d
C
=
{
λ
1
a
1
+
…
+
λ
n
a
n
∣
λ
1
,
…
,
λ
n
≥
0
,
λ
1
,
…
,
λ
n
∈
R
}
⊂
R
d
we consider the monoid
C
∩
L
. By Gordan's lemma this monoid is finetely generated, i.e., there exists a finite set of lattice points
{
x
1
,
…
,
x
m
}
⊂
C
∩
L
such that every lattice point
x
∈
C
∩
L
is an integer conical combination of these points:
x
=
λ
1
x
1
+
…
+
λ
m
x
m
,
λ
1
,
…
,
λ
m
∈
Z
,
λ
1
,
…
,
λ
m
≥
0
The cone C is called pointed, if
x
,
−
x
∈
C
implies
x
=
0
. In this case there exists a unique minimal generating set of the monoid
C
∩
L
- the Hilbert basis of C. It is given by set of irreducible lattice points: An element
x
∈
C
∩
L
is called irreducible if it can not be written as the sum of two non-zero elements, i.e.,
x
=
y
+
z
implies
y
=
0
or
z
=
0
.
