$d$ polynomials, by asking that, for all $s\not= 0$, the total Milnor number $\mu(s)$ of $f_s$ is exactly equal to $(d-1)^n$. Such polynomials $f_s$, which we call [*general-at-infinity*]{}, have also the maximum number of vanishing cycles (Proposition \[p:max\], Corollary \[l:bif\]).

Our study is placed in the framework of polynomials with isolated singularities, including singularities at infinity, in some appropriate sense. We show that in this class, the number of vanishing cycles is upper semi-continuous (Proposition \[l:semicont\]), which is just opposite as in the case of germs.

The loss of vanishing cycles is actually due to the loss of some of the singular points of $f_s$, as $s\to 0$. This can happen in two ways: the corresponding critical value in ${{\mathbb C}}$ of such a singularity tends either to a finite limit or to infinity. This phenomenon produces singularities at infinity for $f_0$, which are quantified by the total Milnor-Lê number $\lambda$. In other words, loss of $\mu$ yields gain of $\lambda$: Theorem \[t:lambdaexchange\] shows how the $\mu 
\rightleftarrows \lambda$ exchange takes place whenever singularities tend to some point at infinity of a compactified fiber. In this way one can detect jumps in the topology by studying the behaviour of the 