find in §\[s:zeta\] several formulas for the zeta-function of the monodromy.

There is a whole section of examples on the $\mu 
\rightleftarrows \lambda$ exchange and the $s$-monodromy. Finally, we explain to what extent our study of a global situation (deformations of polynomials) relates to Arnol’d’s classification of simple germs of meromorphic functions of type $H/Z^d$.

This research originated in a discussion with Vladimir Igorevich Arnol’d in 1999 in Lille, on how vanishing cycles may appear in the list of simple meromorphic germs (see §\[s:arnold\]). Part of it was done as a 2002 RiP program at Oberwolfach, with support from the Volkswagen-Stiftung. The authors benefited from the excellent working conditions offered by the Mathematisches Forschungsinstitut Oberwolfach.

Invariants associated to a polynomial functions {#s:inv}
===============================================

For any polynomial function $f: {{\mathbb C}}^n \to {{\mathbb C}}$, there exists a minimal finite set $B_f\subset {{\mathbb C}}$, called the set of [*atypical values*]{}, such that the restriction $f_|: 
{{\mathbb C}}^n \setminus f^{-1}(B_f) \to {{\mathbb C}}\setminus B_f$ is a locally trivial fibration. If $f$ has isolated singularities at infinity, in the sense of [@ST], then its general fiber $G$ is homotopy equivalent to a bouquet of spheres $\vee_\gamma S^{n-1}$, cf. [*loc. cit*]{}.

In this case, the 