Ju n 19 99 On the connectivity of complex affine hypersurfaces , II

Improving previous results on the connectivity of affine hypersurfaces, the first author has proved the part (ii) of this result in the case of the usual weights w = (1, 1, ..., 1) in [D1]. The proof there was based on the properties of tame and quasi-tame polynomials introduced by Broughton [B], Némethi [N1], [N2] and Némethi-Zaharia [NZ]. More precisely, the general case was reduced to these special classes of polynomials by applying an affine Lefschetz hyperplane section theorem due to Hamm [H]. This proof cannot be applied for arbitrary weights since in general we don’t have enough weighted hyperplanes to proceed by induction (even though Lemma 9. in [D1] can be easily extended to show that the polynomial f is quasitame when dimS(f) = 0).