For a non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of  or , where the case n=2 is solved. As applications, we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of , , ,  and , where the case n=2 is treated.
The main result of the paper provides an explicit description of an embedded resolution of the hypersurface germ (f f(x; y) +z 2 = 0 g; 0)(C 3; 0) for an arbitrary f : (C 2;0)(C; 0), The topological data of the exceptional divisors, their intersections and embeddings is codified in a (two-dimensional) \resolution graph".