Let
F
∈
Z
[
x
0
,
…
,
x
n
]
F \in \mathbb {Z}[x_0, \ldots , x_n]
be homogeneous of degree
d
d
and assume that
F
F
is not a ‘nullform’, i.e., there is an invariant
I
I
of forms of degree
d
d
in
n
+
1
n+1
variables such that
I
(
F
)
≠
0
I(F) \neq 0
. Equivalently,
F
F
is semistable in the sense of Geometric Invariant Theory. Minimizing
F
F
at a prime
p
p
means to produce
T
∈
M
a
t
(
n
+
1
,
Z
)
∩
G
L
(
n
+
1
,
Q
)
T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q})
and
e
∈
Z
≥
0
e \in \mathbb {Z}_{\ge 0}
such that
F
1
=
p
−
e
F
(
[
x
0
,
…
,
x
n
]
⋅
T
)
F_1 = p^{-e} F([x_0, \ldots , x_n] \cdot T)
has integral coefficients and
v
p
(
I
(
F
1
)
)
v_p(I(F_1))
is minimal among all such
F
1
F_1
. Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors
w
∈
Z
≥
0
n
+
1
w \in \mathbb {Z}_{\ge 0}^{n+1}
to
F
F
. We show that for any dimension
n
n
and degree
d
d
, there is a complete set of weight vectors consisting of
[
0
,
w
1
,
w
2
,
…
,
w
n
]
[0,w_1,w_2,\dots ,w_n]
with
0
≤
w
1
≤
w
2
≤
⋯
≤
w
n
≤
2
n
d
n
−
1
0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1}
. When
n
=
2
n = 2
, we improve the bound to
d
d
. This answers a question raised by Kollár. These results are valid in a more general context, replacing
Z
\mathbb {Z}
and
p
p
by a PID
R
R
and a prime element of
R
R
.
Based on this result and a further study of the minimization process in the planar case
n
=
2
n = 2
, we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree
d
d
. We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma.