Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$. Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$. Similarly, define $t_y(E)$ to be the maximum among $0$ and the $y$-degree of $E(0,y)$.
The following nice result appears in several places, see for example,Proposition 2.1 (or Lemma 1.14 or Proposition 10.2.6).
Nice result: Let $A,B \in k[x,y]$ satisfy $\operatorname{Jac}(A,B) \in k-\{0\}$ (such $A,B$ is called a Jacobian pair). Assume that the $(1,1)$-degree of $A$, $\deg(A)$, is $>1$ and the $(1,1)$-degree of $B$, $\deg(B)$, is $>1$. Then the numbers $t_x(A),t_y(A),t_x(B),t_y(B)$ are all positive.
My question: Is the same result holds in the first Weyl algebra over $k$, $A_1(k)$? where instead of the Jacobian we take the commutator.
My answer: Of course, we must first define $t_x(A),t_y(A),t_x(B),t_y(B)$ in $A_1(k)$; it seems to me that the same definition holds for $A_1(k)$, or am I missing something? Perhaps it is not possible to consider $E(x,0)$, where $E \in A_1(k)$?
If I am not wrong, the proof of Proposition 2.1 can be adjusted to the non-commutative case:
(i) It is easy to see that Lemma 1.3 has a non-commutative analog.
(ii) Replacing the Jacobian by the commutator yields a similar result (use $[ab,c]=a[b,c]+[a,c]b$), and then the same conclusion follows.
One has to be careful what exactly is the similar result, since, for example, $[y^3,B]=y^2c+ycy+cy^2$, where $c:=[y,B]$. Then $[y^3,B]=3cy^2+\epsilon$, where $\epsilon \in A_1$ has degree $<\deg(c)+2$. We can consider the highest $(0,1)$-degree terms.
Indeed, suppose that $t_x(A)=0$. Then $A$ is divisible by $y$, so $A=\tilde{A}y^t$, where $t \geq 1$ and $\tilde{A}$ is an element of $A_1$ not divisible by $y$.
Actually, immediately $t=1$, because Lemma 1.3 (= the commutative and its non-commutative analog) says that if $(1,0) \notin \operatorname{Supp}(A)$, then $(0,1) \in \operatorname{Supp}(A)$.
We have, $1=[A,B]=[\tilde{A}y,B]=\tilde{A}[y,B]+[\tilde{A},B]y$.
By considerations of $(1,1)$-degrees, we obtain that $B$ is of the following form: $B=\lambda x + \mu +\tilde{B}y$, for some $\tilde{B} \in A_1$, $\lambda \in k-\{0\}$, $\mu \in k$ (w.l.o.g. $\mu=0$).
In other words, the apriori $B$ of the form $B=w+\tilde{B}y$ with $w \in k[x]$ and $\deg(w) \geq 1$, actually has $\deg(w)=1$.
Hopefully, this further information answers my question in the positive (I have asked a separate question about this further information).
Remark: If I am not wrong, I can prove the following: If the nice result holds in the first Weyl algebra, then the Dixmier Conjecture is true.
I have asked this question here.
Thank you very much!
