affine critical points in the deformation.

Our approach is based on the interplay between the affine singularities of the polynomial and the singularities of its compactified fibers on the hyperplane at infinity $H^{{\infty}}$. Actually there are finitely many one-parameter families of [*boundary singularities*]{} attached to the compactifies fibers of a polynomial function and their restrictions to the hyperplane at infinity. Arnol’d has already remarked in [@Ar2] that there is a close relation between the classification of meromorphic germs of type $H/Z^d$ and the one of germs of boundary singularities [@Ar1].

In a one-parameter deformation $f_s$ of $f_0$ there is a well-defined monodromy of a general fiber of $f_s$ (where $s$ is close enough to $0$) over a small circle $\partial \bar D \times \{ t\}$ centered at $(0,t)\in {{\mathbb C}}\times {{\mathbb C}}$. We call it the [*$s$-monodromy*]{}. Depending on the position of $t$ with respect to the affine discriminant $\Delta_P$ (Definition \[d:gamma\]), there are 3 types of $s$-monodromy, which we call [*generic, atypical*]{}, respectively [*at infinity*]{}. We compare these global monodromies to the monodromies of the germs of boundary singularities of $f_0$. On the other hand we decompose the generic $s$-monodromy according to the types of branches of $\Delta_P$. We 