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Here is the question which could be quit difficult (but could be not): Let $C$ be a field of complex numbers and $f \in C[x,y]$ be a polynomial such that there exist $g \in C[x,y]$ and $Jac(f,g) \in C^{*}$ i.e. determinant of Jacobian matrix of polynomials $f$ and $g$ is a nonzero constant. Question: Is it true that $f$ is irreducible? Any comments are welcome! |
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Edit: The statement in the next paragraph is wrong! I misunderstood the result of Kaliman: it says that given $(f,g)$ as in the question, there is a polynomial automorphism $\phi$ of $\mathbb{C}^2$ such that each fiber of $\phi \circ (f,g): \mathbb{C}^2 \to \mathbb{C}^2$ is irreducible. So I would assume it is still hard to give a positive answer to the question, but clearly what I wrote below is false. I would assume the question is quite difficult, since a positive answer would imply the Jacobian conjecture by this result of Kaliman. |
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