the lines $\{ p_i\} \times {{\mathbb C}}$, where $p_i$ is a singular point of ${{\mathbb X}}_t:= \tau^{-1}(t)$, situated on the part at infinity ${{\mathbb X}}_t^{{{\infty}}}:= \tau^{-1}(t) \cap H^{{{\infty}}}$ (and not depending on $t$).

Let now $(p_i,t)\in {{\mathbb P}}^{n-1}\times {{\mathbb C}}$ be a singular point of ${{\mathbb X}}_t^{{{\infty}}}$. This may be a singular point of ${{\mathbb X}}_t$, or a point where ${{\mathbb X}}_t$ is non-singular but tangent to $H^{{{\infty}}}$ at $p_i$. Let then denote by: $$\mu_{p_i}(t) := \mu({{\mathbb X}}_t, p_i), \ \ \ \mu_{p_i}^{{\infty}}(t) := \mu({{\mathbb X}}_t^{{{\infty}}}, p_i),$$ the Milnor numbers of the respective hypersurface germs at $p_i$. Note that $\mu_{p_i}^{{\infty}}(t)$ is independent of $t$, therefore we shall use the notation $\mu_{p_i}^{{\infty}}$. In contrast, $\mu_{p_i}(t)$ may jump at a finite number of values of $t$. Let then denote by $\mu_{p_i, gen}$ the value of $\mu_{p_i}(t)$ for generic $t$. The jump of $\mu_{p_i}$ at $t$ is measured by the local Milnor-Lê number, cf. [@Pa], [@ST]: $$\label{eq:lambda}
\lambda_{p_i, t} = \mu_{p_i}(t) - \mu_{p_i, gen}.$$ We say that $(p_i,t)$ is a [*singularity at infinity of $f$*]{} if and only if $\lambda_{p_i, t}> 0$. Then the set of atypical values $B_f$ consists of the values $t\in {{\mathbb C}}$ such that either $t$ is a critical 