Let k be a field of characteristic zero and let E=E(x,y) be an element of 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:
Let A,B be two elements of k[x,y] having an invertible Jacobian (= their Jacobian is a non-zero scalar); such A,B is called a Jacobian pair.
Assume that the (1,1)-degree of A is >1 and the (1,1)-degree of B is >1.
Then the numbers t_x(A),t_y(A),t_x(B),t_y(B) are all positive.
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.
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 is an element of A_1(k)?
Thank you very much! Please see https://mathoverflow.net/questions/334897/a-non-commutative-analog-of-a-result-concerning-a-jacobian-pair
(1) If I am not wrong, the proof of Proposition 2.1 of Article On Appelgate-Onishi's Lemmas
can be adjusted to the non-commutative case: (i) It is easy to see that Lemma 1.3 of that paper 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.However, there seem to be a problem with the computation of the commutator ('the similar result' does not imply what we want).
(2) Luckily, the proof of A. van den Essen of Proposition 10.2.6 in his famous book about the Jacobian Conjecture seem to still hold if we replace k[x,y] by A_1(k), isn't it?
(3) BTW, if I am not wrong, this non-commutative analog, together with several not-too-difficult arguments imply that the Dixmier Conjecture is true. B"H, I will later add my proof.
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