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 $$\mathbb{C}[x, y] = \mathbb{C}[f, g]+g\cdot\mathbb{C}[x, y]$$?

• Do you have some reason to suspect this? This is equivalent to the Jacobian conjecture. – Mohan Jul 21 at 17:44
• Do you have proof of its equivalence to JC? – Algorithm Jul 21 at 17:52
• If JC is true, the above equality is obvious. If the above equality is true in general, consider $g_i\in\mathbb{C}[x_1,\ldots, x_{n+1}]$ with $g_i=f_i, i\leq n$ and $g_{n+1}=x_{n+1}$ and apply the equality for the $g_i$s. – Mohan Jul 21 at 18:00
• So You know that if $Q(n)$ is my question and $JC(n)$ is the Jacobian conjecture, then $JC(n)=>Q(n)=>JC(n-1)$. Assume we have proven $JC(i)$ for $i<n$ then can $Q(n)$ be proved? – Algorithm Jul 21 at 18:16
• As I know Jacobian conjecture is open even for $n=2$, so I edited the question to the case $n=2$. – Algorithm Jul 21 at 18:40

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 $$\mathbb{C}[f,g]=\mathbb{C}[x,y]$$.