There is an interesting theorem about the first Weyl algebra $A_1(k)= k \langle x,y | yx-xy= 1 \rangle$, $k$ is a field of characteristic zero, the Polarization Theorem, Corollary 5.5 by A. Joseph.
The Polarization Theorem states that a $k$-endomorphism $\theta$ of $A_1(k)$ is an automorphism or there exists a positive integer $m$ and an injective map $\psi^{(m)}$ such that for some $l\neq 0$, $f_{1,-1}(\psi^{(m)}(\theta(y)))=\beta yx^{1+l/m}$ and $f_{1,-1}(\psi^{(m)}(\theta(x)))=-(m/\beta l) x^{-l/m}$, $\beta \in k-\{0\}$.
Is it known that a similar theorem holds for $k[x,y]$? namely, the following: A $k$-algebra endomorphism $\theta$ of $k[x,y]$ is an automorphism of $k[x,y]$ or there exists a positive integer $m$ and a map $\psi^{(m)}$ having a non-zero scalar Jacobian, $\operatorname{Jac}(\psi^{(m)}(x),\psi^{(m)}(y)) \in k^{\times}$, such that for some $l\neq 0$, $f_{1,-1}(\psi^{(m)}(\theta(y)))=\beta yx^{1+l/m}$ and $f_{1,-1}(\psi^{(m)}(\theta(x)))=-(m/\beta l) x^{-l/m}$, $\beta \in k-\{0\}$.
Joseph's theorem is very difficult to prove, so I guess that its commutative analog should also be difficult to prove.
I thought to apply the implication $DC(2) \rightarrow JC(2)$, but we do not have Joseph's theorem in $A_2(k)$. Or perhaps to apply $JC(2) \rightarrow DC(1)$.
Thank you very much!
