value of $f$ or $(p_i,t)$ is a singularity at infinity of $f$ for some $p_i\in H^{{\infty}}$. The total Milnor-Lê number of $f$ is the sum of the local ones: $$\label{eq:lambdatotal}
 \lambda = \sum_{t\in B_f}\sum_i \lambda_{p_i, t} =  \sum_{t\in B_f}\sum_i(\mu_{p_i}(t) - \mu_{p_i, gen}).$$ Let us compute the $(n-1)$-th betti number of the fibers $F_t = f^{-1}(t)$, using information on the singularities of their compactifications ${{\mathbb X}}_t$ and of their restrictions to the hyperplane at infinity ${{\mathbb X}}_t^{{{\infty}}}$. Recall that the Euler characteristic of the general fiber of a generic polynomial of degree $d$ is $\chi(n,d) = 1+(-1)^{n-1}(d-1)^n$. Comparing it to the Euler characteristic of $F_t$, one obtains the following formula, see e.g. [@Di]: $$\chi(n,d) - \chi(F_t) = (-1)^{n-1}\sum_i(\mu_{p_i}(t) + \mu_{p_i}^{{\infty}}).$$ This is valid for all $t$, as long as $F_t$ is non-singular. When $F_t$ contains affine critical points, then the formula has to be corrected by adding to the right hand side the local Milnor numbers of these singularities. When $F_t$ is the general fiber $G$, we obtain:

$$\label{eq:chi}
  b_{n-1}(G) = (d-1)^n - \sum_i(\mu_{p_i, gen} + \mu_{p_i}^{{\infty}}).$$

We have now two different formulas, (\[eq:mulambda\]) and (\[eq:chi\]), both computing the number of vanishing cycles for a [[$\mathcal F$[[ISI]{}]{}]{}]{} polynomial.

Deformations of [[$\mathcal 