Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$.
Is it true that $\mathbb{C}[x, y] = \mathbb{C}[f, g]+g\cdot\mathbb{C}[x, y]$?
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Sign up to join this communityAssume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$.
Is it true that $\mathbb{C}[x, y] = \mathbb{C}[f, g]+g\cdot\mathbb{C}[x, y]$?
This is still is equivalent to JC.
Your equality says, $\mathbb{C}[x,y]=\mathbb{C}[f,g]+g\mathbb{C}[x,y]$, the last term is equal to $\mathbb{C}[f]+g\mathbb{C}[f,g]+g\mathbb{C}[x,y]=\mathbb{C}[f]+g\mathbb{C}[x,y]$, since $g\mathbb{C}[f,g]\subset g\mathbb{C}[x,y]$. This says, the map $\mathbb{C}[f]\to \mathbb{C}[x,y]/g\mathbb{C}[x,y]$ is onto and then it is clear that this is an isomorphism. Then, $g=0$ is an embedded line in $\mathbb{C}^2$ and by Abhyankar-Moh, is a co-ordinate line after an automorphism. Then, it is easy to verify that $\mathbb{C}[f,g]=\mathbb{C}[x,y]$.