On the topology of a generic fibre of a polynomial function

Abstract

In this work we study the topologies of the fibres of some families of complex polynomial functions with isolated critical points. We consider polynomials with some transversality conditions at infinity and compute explicitly its global Milnor number μ(f). the invariant λ(f) and therefore the Euler characteristic of its generic fibre. We show that under some mild ransversality condition (transversal at infinity) the behavior of f at infinity is good and the topology of the generic fibre is determined by the two homogeneous parts of higher degree of f Finally we study families of polynomials, called two-term polynomials. This polynomials may have atypical values at infinity. Given such a two-term polynomial f we characterize its atypical values by some invariants of f. These polynomials are a source of interesting examples.