Nontrivial Rational Polynomials in Two Variables Have Reducible Fibres

We shall call a polynomial map f : C → C a “coordinate” if there is a g such that (f, g) : C → C is a polynomial automorphism. Equivalently, by AbhyankarMoh and Suzuki, f has one and therefore all fibres isomorphic to C. Following [7] we call a polynomial f : C → C “rational” if the general fibres of f (and hence all fibres of f) are rational curves. The following theorem, which says that a rational polynomial map with irreducible fibres cannot be part of a counterexample to the 2-dimensional Jacobian Conjecture, has appeared in the literature several times. It appears with an algebraic proof in Razar [12]. It is Theorem 2.5 of Heitmann [4] (as corrected in the Corrigendum), and Lê and Weber, who give a geometric proof in [6], also cite the reference Friedland [3], which we have not seen.