In der algebraischen Geometrie gibt die Hilbert-Funktion Informationen über die Anzahl der Hyperflächen zu einem gegebenen Grad. Für hinreichend große Argumente stimmt sie mit einem als Hilbert-Polynom bezeichneten Polynom überein.
Sei
X
⊂
P
n
{\displaystyle X\subset P^{n}}
projektive Varietät mit Verschwindungsideal
I
(
X
)
⊂
K
[
Z
0
,
…
,
Z
n
]
{\displaystyle I(X)\subset K\left[Z_{0},\ldots ,Z_{n}\right]}
Für
d
≥
1
{\displaystyle d\geq 1}
I
(
X
)
d
=
I
(
X
)
∩
K
[
Z
0
,
…
,
Z
n
]
d
{\displaystyle I(X)_{d}=I(X)\cap K\left[Z_{0},\ldots ,Z_{n}\right]_{d}}
der homogene Anteil vom Grad
d
{\displaystyle d}
Koordinatenring
S
(
X
)
{\displaystyle S(X)}
graduierter Ring
S
(
X
)
=
⨁
d
≥
0
S
(
X
)
d
{\displaystyle S(X)=\bigoplus _{d\geq 0}S(X)_{d}}
mit
S
(
X
)
d
=
K
[
Z
0
,
…
,
Z
n
]
d
/
I
(
X
)
d
{\displaystyle S(X)_{d}=K\left[Z_{0},\ldots ,Z_{n}\right]_{d}/I(X)_{d}}
Die Dimension von
I
(
X
)
d
{\displaystyle I(X)_{d}}
X
{\displaystyle X}
d
{\displaystyle d}
Hilbert-Funktion
h
X
{\displaystyle h_{X}}
h
X
(
d
)
=
dim
S
(
X
)
d
{\displaystyle h_{X}(d)=\dim S(X)_{d}}
sie gibt also die Kodimension von
I
(
X
)
d
{\displaystyle I(X)_{d}}
Sei
X
=
{
[
1
:
0
:
0
]
,
[
0
:
1
:
0
]
,
[
0
:
0
:
1
]
}
⊂
P
2
{\displaystyle X=\left\{\left[1\colon 0\colon 0\right],\left[0\colon 1\colon 0\right],\left[0\colon 0\colon 1\right]\right\}\subset P^{2}}
h
X
(
d
)
=
3
{\displaystyle h_{X}(d)=3}
d
≥
1
{\displaystyle d\geq 1}
Sei
X
=
{
[
1
:
0
:
0
]
,
[
0
:
1
:
0
]
,
[
1
:
1
:
0
]
}
⊂
P
2
{\displaystyle X=\left\{\left[1\colon 0\colon 0\right],\left[0\colon 1\colon 0\right],\left[1\colon 1\colon 0\right]\right\}\subset P^{2}}
h
X
(
1
)
=
2
{\displaystyle h_{X}(1)=2}
h
X
(
d
)
=
3
{\displaystyle h_{X}(d)=3}
d
≥
2
{\displaystyle d\geq 2}
Sei
X
⊂
P
n
{\displaystyle X\subset P^{n}}
m
{\displaystyle m}
h
X
(
d
)
=
m
{\displaystyle h_{X}(d)=m}
d
≥
m
−
1
{\displaystyle d\geq m-1}
Sei
X
⊂
P
2
{\displaystyle X\subset P^{2}}
k
{\displaystyle k}
Kurve . Dann ist
h
X
(
d
)
=
d
⋅
k
−
1
2
k
(
k
−
3
)
{\displaystyle h_{X}(d)=d\cdot k-{\frac {1}{2}}k(k-3)}
d
≥
k
{\displaystyle d\geq k}
D. Eisenbud : Commutative algebra. With a view toward algebraic geometry , Graduate Texts in Mathematics 150, Springer-Verlag New York, ISBN 0-387-94268-8 
![{\displaystyle I(X)\subset K\left[Z_{0},\ldots ,Z_{n}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16d672978e565f94fc9993c793d1fdeabbc41f1)
![{\displaystyle I(X)_{d}=I(X)\cap K\left[Z_{0},\ldots ,Z_{n}\right]_{d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c96248f7a73e304af803f8d70a2816215a30ffdb)
![{\displaystyle S(X)_{d}=K\left[Z_{0},\ldots ,Z_{n}\right]_{d}/I(X)_{d}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed7c8253bba3efe7e7be843f7967b7052213577)
![{\displaystyle X=\left\{\left[1\colon 0\colon 0\right],\left[0\colon 1\colon 0\right],\left[0\colon 0\colon 1\right]\right\}\subset P^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03fbdc9add4da9ec867028b408682d42aeeeec40)
![{\displaystyle X=\left\{\left[1\colon 0\colon 0\right],\left[0\colon 1\colon 0\right],\left[1\colon 1\colon 0\right]\right\}\subset P^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2d6944f9d3c258c42f08b096fd6cf9d3775478)
![{\displaystyle H_{X}\in \mathbb {Q} \left[d\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1578ca79a38793ad7ceef5c415295db5349b4a5)
