Prove for the polynomial ring
K
[
X
1
,
…
,
X
n
]
{\displaystyle {}K[X_{1},\ldots ,X_{n}]}
K
{\displaystyle {}K}
Show that the ring of differential operators on
K
[
X
1
,
…
,
X
n
]
{\displaystyle {}K[X_{1},\ldots ,X_{n}]}
Let
H
∈
K
[
X
1
,
…
,
X
n
]
{\displaystyle {}H\in K[X_{1},\ldots ,X_{n}]}
e
{\displaystyle {}e}
e
H
=
X
1
∂
H
∂
X
1
+
⋯
+
X
n
∂
H
∂
X
n
.
{\displaystyle {}eH=X_{1}{\frac {\partial H}{\partial X_{1}}}+\cdots +X_{n}{\frac {\partial H}{\partial X_{n}}}\,.}
Let
A
{\displaystyle {}A}
R
{\displaystyle {}R}
R
{\displaystyle {}R}
μ
f
:
A
⟶
A
,
x
⟼
f
x
,
{\displaystyle \mu _{f}\colon A\longrightarrow A,x\longmapsto fx,}
denote the
R
{\displaystyle {}R}
f
∈
A
{\displaystyle {}f\in A}
R
{\displaystyle {}R}
φ
1
,
φ
2
:
A
⟶
A
{\displaystyle \varphi _{1},\varphi _{2}\colon A\longrightarrow A}
set
[
φ
1
,
φ
2
]
=
φ
1
∘
φ
2
−
φ
2
∘
φ
1
.
{\displaystyle {}[\varphi _{1},\varphi _{2}]=\varphi _{1}\circ \varphi _{2}-\varphi _{2}\circ \varphi _{1}\,.}
Suppose that a
R
{\displaystyle {}R}
δ
:
A
→
A
{\displaystyle {}\delta \colon A\rightarrow A}
g
∈
A
{\displaystyle {}g\in A}
[
δ
,
μ
g
]
{\displaystyle {}[\delta ,\mu _{g}]}
Let
R
{\displaystyle {}R}
K
{\displaystyle {}K}
E
:
R
→
R
{\displaystyle {}E\colon R\rightarrow R}
K
{\displaystyle {}K}
E
{\displaystyle {}E}
≤
n
{\displaystyle {}\leq n}
For arbitrary elements
f
0
,
f
1
,
…
,
f
n
∈
R
{\displaystyle {}f_{0},f_{1},\ldots ,f_{n}\in R}
E
(
f
0
⋯
f
n
)
=
∑
s
=
1
n
(
−
1
)
s
+
1
∑
I
⊆
{
0
,
…
,
n
}
,
#
(
I
)
=
s
∏
i
∈
I
f
i
⋅
E
(
∏
i
∉
I
f
i
)
=
∑
I
⊆
{
0
,
…
,
n
}
,
I
≠
∅
(
−
1
)
#
(
I
)
+
1
∏
i
∈
I
f
i
⋅
E
(
∏
i
∉
I
f
i
)
.
{\displaystyle {}{\begin{aligned}E(f_{0}\cdots f_{n})&=\sum _{s=1}^{n}(-1)^{s+1}\sum _{I\subseteq \{0,\ldots ,n\},\,{\#\left(I\right)}=s}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}\\&=\sum _{I\subseteq \{0,\ldots ,n\},\,I\neq \emptyset }(-1)^{{\#\left(I\right)}+1}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}.\end{aligned}}}
Recall the implicite function theorem.
Describe the derivations on
K
[
X
,
Y
,
Z
]
/
(
Z
2
−
X
2
−
Y
2
)
{\displaystyle {}K[X,Y,Z]/(Z^{2}-X^{2}-Y^{2})}
We consider the twodimensional cone
C
{\displaystyle {}C}
R
≥
0
(
7
2
)
{\displaystyle {}\mathbb {R} _{\geq 0}{\begin{pmatrix}7\\2\end{pmatrix}}}
R
≥
0
(
5
6
)
{\displaystyle {}\mathbb {R} _{\geq 0}{\begin{pmatrix}5\\6\end{pmatrix}}}
C
∩
Z
2
{\displaystyle {}C\cap \mathbb {Z} ^{2}}
Let
C
⊆
R
n
{\displaystyle {}C\subseteq \mathbb {R} ^{n}}
F
{\displaystyle {}F}
ℓ
:
R
n
⟶
R
{\displaystyle \ell \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} }
be a linear form, such that its kernel contains the facet. Suppose that the linear form is given by integers which are coprime. Show that
ℓ
{\displaystyle {}\ell }
−
ℓ
{\displaystyle {}-\ell }
F
{\displaystyle {}F}
Determine for the monoid ring
K
[
X
,
Y
,
Z
]
/
(
Z
2
−
X
Y
)
{\displaystyle {}K[X,Y,Z]/(Z^{2}-XY)}
X
{\displaystyle {}X}
Z
{\displaystyle {}Z}
X
Y
{\displaystyle {}XY}
Determine for the monoid ring
K
[
X
,
Y
,
Z
]
/
(
Z
W
−
X
Y
)
{\displaystyle {}K[X,Y,Z]/(ZW-XY)}
X
{\displaystyle {}X}
X
Y
{\displaystyle {}XY}
X
2
{\displaystyle {}X^{2}}
Determine for the numerical semigroup ring
K
[
U
3
,
U
4
,
U
5
,
…
]
{\displaystyle {}K[U^{3},U^{4},U^{5},\ldots ]}
U
3
,
U
4
,
U
5
{\displaystyle {}U^{3},U^{4},U^{5}}
![{\displaystyle {}K[X_{1},\ldots ,X_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96607693c80a38f197b1443eb366df0a6bbef1c3)
![{\displaystyle {}H\in K[X_{1},\ldots ,X_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e71c4a14141a395914b02684535f7c5f137054)
![{\displaystyle {}[\varphi _{1},\varphi _{2}]=\varphi _{1}\circ \varphi _{2}-\varphi _{2}\circ \varphi _{1}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7307941213194905b6c44144d9513a4ea67850b3)
![{\displaystyle {}[\delta ,\mu _{g}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5bd0329f0a6f24937bddc092edbdc7758a6d9f)
![{\displaystyle {}K[X,Y,Z]/(Z^{2}-X^{2}-Y^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd91abcb939570e3785c41cf0b44f2a928e7d1e)
![{\displaystyle {}K[X,Y,Z]/(Z^{2}-XY)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41e72a3fceec6ed43d342271f7b1c3a1b91e824f)
![{\displaystyle {}K[X,Y,Z]/(ZW-XY)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7860f5961ec30fb2306d2e57d921907b950e16)
![{\displaystyle {}K[U^{3},U^{4},U^{5},\ldots ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7aee23977c89d27cdbba93e9bfa3a0de2e7fb0)
