Let $k$ be a field of characteristic zero, and let $k[x,y]$ be the polynomial ring in two variables $x$ and $y$. Let $a,b,R \in k[x,y]$. By definition, $\operatorname{Jac}(R,a):=R_xa_y-R_ya_x$.
Assume that $\operatorname{Jac}(R,a)=\lambda R^mb$, for some $\lambda \in k-\{0\}$ and $m \in \mathbb{N}$.
Is it true that $b \in k-\{0\}$? Or can one find an easy counterexample?
Example for $\lambda=b=1$ and $m=2$: Take $R=x^2y^3$ and $a=-x^3y^4$. Then, $R_x=2xy^3$, $R_y=3x^2y^2$, $a_x=-3x^2y^4$, $a_y=-4x^3y^3$.
Therefore, $\operatorname{Jac}(R,a)=(2xy^3)(-4x^3y^3)-(3x^2y^2)(-3x^2y^4)= -8x^4y^6+9x^4y^6=x^4y^6=R^2$.
Any hints and comments are welcome!