Hier berechnen wir das asymptotische Verhalten der symmetrischen Potenzen sowie der Frobenius-Pullbacks von
S
y
z
2
{\displaystyle {}Syz_{2}}
R
=
k
[
x
,
y
,
z
,
w
]
/
(
x
2
+
y
2
+
z
2
+
w
2
)
{\displaystyle {}R=k[x,y,z,w]/(x^{2}+y^{2}+z^{2}+w^{2})}
k
=
Q
{\displaystyle {}k=\mathbb {Q} }
k
=
F
p
{\displaystyle {}k=\mathbb {F} _{p}}
p
>
q
{\displaystyle {}p>q}
k
=
F
p
{\displaystyle {}k=\mathbb {F} _{p}}
q
=
p
e
{\displaystyle {}q=p^{e}}
Wir erwarten, dass die normierten Werte jeweils gegen 4,6666... konvergieren, siehe hier .
q
k
{\displaystyle {}k}
∑
h
2
(
S
q
(
S
y
z
2
)
(
m
)
)
{\displaystyle {}\sum h^{2}(S^{q}(Syz_{2})(m))}
Rang von
S
q
(
S
y
z
2
)
{\displaystyle {}S^{q}(Syz_{2})}
durch Rang
q
3
{\displaystyle {}q^{3}}
k
{\displaystyle {}k}
∑
h
2
(
F
e
(
S
y
z
2
)
(
m
)
)
{\displaystyle {}\sum h^{2}(F^{e}(Syz_{2})(m))}
Rang von F^e(Syz_2)
durch Rang
q
3
{\displaystyle {}q^{3}}
1
Q
{\displaystyle {}\mathbb {Q} }
F
11
{\displaystyle {}\mathbb {F} _{11}}
8
4
2
2
Q
{\displaystyle {}\mathbb {Q} }
F
11
{\displaystyle {}\mathbb {F} _{11}}
276
10
3,45
3
Q
{\displaystyle {}\mathbb {Q} }
F
11
{\displaystyle {}\mathbb {F} _{11}}
2144
20
3,9703...
F
3
{\displaystyle {}\mathbb {F} _{3}}
400
4
3,7037...
4
F
11
{\displaystyle {}\mathbb {F} _{11}}
9501
35
4,2415...
5
F
7
{\displaystyle {}\mathbb {F} _{7}}
30848
56
4,4068...
F
5
{\displaystyle {}\mathbb {F} _{5}}
2040
4
4,08
6
F
7
{\displaystyle {}\mathbb {F} _{7}}
81378
84
4,4851...
7
F
11
{\displaystyle {}\mathbb {F} _{11}}
120
F
7
{\displaystyle {}\mathbb {F} _{7}}
5824
4
4,2448...
8
9
F
3
{\displaystyle {}\mathbb {F} _{3}}
12648
4
4,3374...
![{\displaystyle {}R=k[x,y,z,w]/(x^{2}+y^{2}+z^{2}+w^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b268f596a9bc3e05e6c15adc077e507852cbdc46)
