Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$ be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
So in $R_{-1}$ we have: $xx^{-1}=x^{-1}x=1$.
Recall that in $k[x,y]$, $p,q \in k[x,y]$ are $k$-algebraically independent if and only if $\operatorname{Jac}(p,q) \neq 0$.
Can we characterize two $k$-algebraically independent elements $\{P,Q\}$ of $R_{-1}$ in a similar way that we characterize them in $k[x,y]$?
(Perhaps here we cannot expect an "if and only if" statement, only one direction statement).
Edit: More generally, let $m$ be a positive integer and let $R_{-1/m}:=k[x^{1/m},x^{-1/m},y]$ be the $k$-algebra of polynomials in variables $x^{1/m},x^{-1/m},y$, subject to the relation $(x^{-1/m})^m x= x (x^{-1/m})^m= 1$.
Can we characterize two $k$-algebraically independent elements $\{P,Q\}$ of $R_{-1/m}$ in a similar way that we characterize them in $k[x,y]$?
Now we should work in the field $k(x^{1/m},x^{-1/m},y)$; is the known result mentioned in the comments also valid here?
It seems to me that the answer is positive, but I am not sure how to prove it. Perhaps one can adjust the arguments in $k[x,y]$ to arguments in $k[x^{1/m},x^{-1/m},y]$?
Any hints and comments are welcome!
Thank you very much!