Irregular fibers of complex polynomials in two variables

Let f : C −→ C be a polynomial. The bifurcation set B for f is the minimal set of points of C such that f : C \f−1(B) −→ C\B is a locally trivial fibration. For c ∈ C, we denote the fiber f−1(c) by Fc. The fiber Fc is irregular if c is in B. If s / ∈ B, then Fs is a generic fiber and is denoted by Fgen . The tube Tc for the value c is a neighborhood f −1(D2 ε(c)) of the fiber Fc, where D 2 ε(c) stands for a 2-disk in C, centered at c, of radius ε ≪ 1. We assume that affine critical singularities are isolated. The value c is regular at infinity if there exists a compact set K of C2 such that the restriction of f to Tc \K −→ D 2 ε(c) is a locally trivial fibration. Set n = 2. Let jc : H1(Fc) −→ H1(Tc) be the morphism induced by the inclusion of Fc in Tc. One of the consequences of our study of jc is the following :