Assume that the algebraically independent polynomials $$f, ginmathbb{C}[x, y]$$ are such that the Jacobian matrix $$text{Jac}_{x, y}^{f, g}inmathbb{C}setminus{0}$$.

Is it true that $$mathbb{C}[x, y] = mathbb{C}[f, g]+gcdotmathbb{C}[x, y]$$?

MathOverflow Asked on November 9, 2021

This is still is equivalent to JC.

Your equality says, $$mathbb{C}[x,y]=mathbb{C}[f,g]+gmathbb{C}[x,y]$$, the last term is equal to $$mathbb{C}[f]+gmathbb{C}[f,g]+gmathbb{C}[x,y]=mathbb{C}[f]+gmathbb{C}[x,y]$$, since $$gmathbb{C}[f,g]subset gmathbb{C}[x,y]$$. This says, the map $$mathbb{C}[f]to mathbb{C}[x,y]/gmathbb{C}[x,y]$$ 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 $$mathbb{C}[f,g]=mathbb{C}[x,y]$$.

Answered by Mohan on November 9, 2021

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