vanishing cycles are quantified by two well-defined, non-negative integers [@ST]: $$\begin{array}{ll} \mu= \mbox{ the total Milnor number  of the affine singularities},\\
 \lambda= \mbox{ the total Milnor-L\^e number at infinity}
 \end{array}$$ and one has: $$\label{eq:mulambda}
 b_{n-1}(G) = \mu + \lambda.$$

In this paper we consider deformations of polynomials within the following class:

\[d:fisi\] We say that $f$ has [*fiberwise isolated singularities at infinity*]{} (for short, $f$ is a [[$\mathcal F$[[ISI]{}]{}]{}]{}  polynomial) if the projective closure of any fiber of $f$ and its slice by the hyperplane at infinity $H^{{\infty}}$ have at most isolated singularities.

This class is large enough to include all reduced polynomial functions in 2 variables. It is a subclass of polynomials having isolated singularities at infinity, in the sense of [@ST]. So the numbers $\mu$ and $\lambda$ are well-defined. The singularities at infinity of a [[$\mathcal F$[[ISI]{}]{}]{}]{} polynomial are detected as follows.

Let ${{\mathbb X}}= \{ ([x;x_0],t)\in {{\mathbb P}}^n\times {{\mathbb C}}\mid \tilde f - tx^d_0 = 0\}$ and let $\tau : {{\mathbb X}}\to {{\mathbb C}}$ be the projection to the $t$-coordinate, where $d= \deg f$ and $\tilde f$ is the degree $d$ homogenisation of $f$ with the variable $x_0$. Then the singular locus of ${{\mathbb X}}$ consists of 