Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$
Let $r \in \mathbb{N}-\{0\}$.
Commutative case: Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
(ii) $l_{1,-1}(p)=\lambda x^{\frac{n}{r}+1}y$ and $l_{1,-1}(q)=\mu x^{-\frac{n}{r}}$, where $n \in \mathbb{Z}-\{0\}$ and $\lambda,\mu \in \mathbb{C}-\{0\}$.
Then we can write $p=\lambda x^{\frac{n}{r}+1}y + A$ and $q=\mu x^{-\frac{n}{r}}+B$, where $v_{1,-1}(A) < \frac{n}{r}$ and $v_{1,-1}(B) < -\frac{n}{r}$.
Question 1: Is necessarily $A \in k[x^{1/r},x^{-1/r}]$ and $B=0$?
Non-commutative case:
Let $A_1(\mathbb{C})$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$. Let $A_1(\mathbb{C})^{-\frac{1}{r}}$ be a ring extension of the first Weyl algebra, generated by $x^{\frac{1}{r}}$, $x^{-\frac{1}{r}}$ and $y$ subject to the relations $yx-xy=1$, $x (x^{-\frac{1}{r}})^r= (x^{-\frac{1}{r}})^r x=1$, and $[y,x^{-\frac{1}{r}}]:=-\frac{1}{r} x^{-\frac{1}{r}-1}$.
Let $f : (x,y) \mapsto (p,q)$ be a map from $A_1(\mathbb{C})$ to $A_1(\mathbb{C})^{-\frac{1}{r}}$ satisfying the following two conditions:
(i) $[q,p]:=qp-pq \in \mathbb{C}-\{0\}$.
Remark: A special case of the above question can be found here (actually, the later edit of that question is the same as the above question).
Thank you veru much!
ac.commutative-algebra polynomials noncommutative-algebra weyl-algebra
Browse other questions tagged ac.commutative-algebra polynomials noncommutative-algebra weyl-algebra or ask your own question.
Does the Polarization Theorem for $A_1(k)$ has an analogue for $k[x,y]$?
A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero
Is $x \in A_1$ left algebraic over the subalgebra generated by $p$ and $q$, $[q,p]=1$?
Invertibility under base change for the Weyl algebra instead of for the polynomial algebra
Solutions to $w_x=CA_x$, $w_y=CA_y$ other than $w=f(A)$ and $C=f'(A)$?
A variation on Dixmier's counterexample concerning centralizers in $A_1$
A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$
Concerning certain Keller maps of $k[x,y]$
A non-commutative analog of a result concerning a Jacobian pair
Is a specific endomorphism of $A_1$ an automorphism?