Let $f,g \in \mathbb{C}[x,y]$.
There is a well-known result, that can be found for example 
[here](http://www.math.wayne.edu/~lml/lmlnotes.pdf), pages 19-20, that says the following:



> 
> $f,g$ are algebraically dependent over $\mathbb{C}$ if and only if their Jacobian $Jac(f,g):=f\_xg\_y-f\_yg\_x$ is zero. 
> 
> 
> 


Actually, this result is valid for $f\_1,\ldots,f\_n \in \mathbb{C}[x\_1,\ldots,x\_n]$, any $n \in \mathbb{N}$.


Now, let $f,g \in A\_1(\mathbb{C})$, where $A\_1(\mathbb{C})$
is the [first Weyl algebra](https://en.wikipedia.org/wiki/Weyl_algebra) over $\mathbb{C}$, namely, the $\mathbb{C}$-algebra
generated by $x,y$ such that $[y,x]=yx-xy=1$.


I wonder if there exists an analog result to the above in $A\_1(\mathbb{C})$,
namely:



> 
> $f,g \in A\_1(\mathbb{C})$ are 'algebraically dependent' over $\mathbb{C}$ if and only if $[f,g]=0$.
> 
> 
> 


One has to be careful because:


(1) One has to define algebraic dependence over $\mathbb{C}$ of two non-commuting elements $f$ and $g$.
Should it be $\sum \lambda\_{ij}f^ig^j=0$, with $\lambda\_{ij} \in \mathbb{C}$ not all zero,
or $\sum \lambda\_{ij}f^ig^j + \sum \mu\_{ij}g^if^j=0$, with $\lambda\_{ij},
\mu\_{ij} \in \mathbb{C}$ not all zero?.
(Perhaps the first definition should be called 'one-sided algebraic dependence',
while the second definition should be called 'two-sided algebraic dependence').


(2) Perhaps [this question](https://math.stackexchange.com/questions/210438/commuting-elements-in-weyl-algebras-of-characteristic-zero) is relevant. The example there (of Dixmier) is of $U,V \in A\_1$, $[U,V]=0$, and $U^3-V^2+1=0$, so it does not contradict my plausible analog result, since those $U$ and $V$ are algebraically dependent.


((3) I am not sure if this is relevant, but in the [above mentioned reference](http://www.math.wayne.edu/~lml/lmlnotes.pdf), on page 11, the [Gelfand-Kirillov dimension](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Kirillov_dimension) is mentioned with connection to transcendence degree; for $\mathbb{C}[x,y]$ those notions coincide. Is the fact that the Gelfand-Kirillov dimension of $A\_1(\mathbb{C})$ is two relevant to my question?).


Thank you very much for any help!



A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero
The reverse implication is true in a considerably more general setting (Burchnall-Chaundy theory). Namely, for any pair $(U,V)$ of commuting meromorphic coefficient differential operators in one variable of order at least one, there is a two-variable polynomial $P(z,w)$ such that $P(U,V)=0$ (the polynomial evaluation is unambiguous because $U$ and $V$ commute). There is an enormous body of literature on this topic related to integrable systems. 


The proposed forward implication does not appear interesting or meaningful to me. (*However, see a very interesting formulation proposed by David Speyer in the comments.*) For example, the defining relation $yx-xy-1=0$ between $x$ and $y$ may be viewed as a form of "noncommutative algebraic dependence", yet of course $x$ and $y$ do not commute. And the proposed "one-sided algebraic dependence" need not have good formal properties, such as symmetry or transitivity. At best, one can attempt to extract such properties from commutativity.


