Polynomial maps of complex plane with branched valued set isomorphic to complex line

We present a completed list of polynomial dominanting maps of C2 with branched value curve isomorphic to the complex line C, up to polynomial automorphisms. AMS Classification : 14 E20, 14 B25. 1. Let f : C −→ C be a polynomial dominanting map, Close(f(C)) = C, and denote by degf the geometric degree of f the number of solutions of the equation f = a for generic points a ∈ C. The branched value set Ef of f is defined to be the smallest subset of C such that the map f : C \ f(Ef ) −→ C n \ Ef (∗) gives a unbranched degf −sheeted covering. It is well-known (see [M]) that the branched value set Ef is either empty set or an algebraic hypersurface and Ef = {a ∈ C n : #f(a) 6= degf}. If Ef = ∅, then f is injective, and hence, f is an automorphism of C by the well-known fact that injective polynomial maps of C are automorphisms (see [R]). The famous Jacobian conjecture ([BCW]) asserts that f must have a singularity if Ef 6= ∅. The question naturally raises as what can be said about polynomial dominanting ∗Supported in part by the National Basic program on Natural Science, Vietnam