Nontrivial rational polynomials in two variables have reducible fibres

Abstract

It has been proved several times in the literature that a polynomial map from $C^2$ to $C$ with irreducible rational fibers cannot be a component of a counterexample to the Jacobian Conjecture. This note points out that this result is empty: it is implicit in 1980 work of Miyanish and Sugie that such a polynomial is equivalent to $f(x,y)=x$ by a polynomial automorphism of $C^2$.

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