# Algebraic differential operators/Introduction/Exercise sheet
### Exercise

Prove for the polynomial ring $K[X_{1},\ldots ,X_{n}]$ over an arbitrary field $K$ that the formal partial derivatives commute.

### Exercise

Show that the ring of differential operators on $K[X_{1},\ldots ,X_{n}]$ is not commutative.

### Exercise

Let $H\in K[X_{1},\ldots ,X_{n}]$ be a homogeneous polynomial of degree $e$. Show the equality
    $eH=X_{1}{\frac {\partial H}{\partial X_{1}}}+\cdots +X_{n}{\frac {\partial H}{\partial X_{n}}}\,.$

### Exercise

Let $A$ be a commutative $R$-algebra over a commutative ring $R$. Let
    $\mu _{f}\colon A\longrightarrow A,x\longmapsto fx,$
denote the $R$-linear multiplication map for $f\in A$. For $R$-linear maps
    $\varphi _{1},\varphi _{2}\colon A\longrightarrow A$
set
    $[\varphi _{1},\varphi _{2}]=\varphi _{1}\circ \varphi _{2}-\varphi _{2}\circ \varphi _{1}\,.$
Suppose that a $R$-derivation $\delta \colon A\rightarrow A$ is given. Show that for all $g\in A$ the map $[\delta ,\mu _{g}]$ is multiplication by some element.

### Exercise

$W\subseteq R$ denote a multiplicative system. Let $D\colon R\rightarrow R$ denote a $K$-derivation. Show that we get via
    $D{\left({\frac {f}{g}}\right)}:={\frac {gD(f)-fD(g)}{g^{2}}}\,$
a derivation on the localization $R_{W}$ which extends $D$.

### Exercise

Let $R$ denote a commutative $K$-algebra and let $E\colon R\rightarrow R$ denote a $K$-linear map. Show that the following statements are equivalent.
* $E$ is a differential operator of order $\leq n$.
* For arbitrary elements $f_{0},f_{1},\ldots ,f_{n}\in R$ we have
    ${\begin{aligned}E(f_{0}\cdots f_{n})&=\sum _{s=1}^{n}(-1)^{s+1}\sum _{I\subseteq \{0,\ldots ,n\},\,{\#\left(I\right)}=s}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}\\&=\sum _{I\subseteq \{0,\ldots ,n\},\,I\neq \emptyset }(-1)^{{\#\left(I\right)}+1}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}.\end{aligned}}$

### Exercise

Recall the implicite function theorem.

### Exercise

Describe the derivations on $K[X,Y,Z]/(Z^{2}-X^{2}-Y^{2})$ and show that there are no unitary derivations on it.

### Exercise

We consider the twodimensional cone $C$ given by the edges $\mathbb {R} _{\geq 0}{\begin{pmatrix}7\\2\end{pmatrix}}$ and $\mathbb {R} _{\geq 0}{\begin{pmatrix}5\\6\end{pmatrix}}$ and the corresponding monoid $C\cap \mathbb {Z} ^{2}$. Determine the describing integral linear forms and the signatures of the cone.

### Exercise

Let $C\subseteq \mathbb {R} ^{n}$ denote a positve rational polyhedrial cone and $F$ a facet of the cone. Let
    $\ell \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R}$
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 $\ell$ or $-\ell$ is the canonical integral linear form of $F$.

### Exercise

Determine for the monoid ring $K[X,Y,Z]/(Z^{2}-XY)$ the canonical unitary differential operators (and their order) for the monomials
* $X$,
* $Z$,
* $XY$.

### Exercise

Determine for the monoid ring $K[X,Y,Z]/(ZW-XY)$ the canonical unitary differential operators (and their order) for the monomials
* $X$,
* $XY$,
* $X^{2}$.

### Exercise

Determine for the numerical semigroup ring $K[U^{3},U^{4},U^{5},\ldots ]$ unitary differential operators for the elements $U^{3},U^{4},U^{5}$.