hyperplane at infinity $H^{{\infty}}$. Those are exactly the points such that $p\in W_s\setminus \Sigma_s$. Notice that $W_s$ is the singular locus of ${{\mathbb Y}}^{{\infty}}_{s,t}$ (and does not depend on $t$). Let us set the following notations, which we shall frequently use: $$\phi(x) := \frac{\partial P}{\partial s}(x,0), \ \  \phi_d := \mbox{ degree } d \mbox{ 
part of } \phi .$$

\[l:phi\] Let $P$ be any deformation of a [[$\mathcal F$[[ISI]{}]{}]{}]{} polynomial $f_0$. Then:

1.  ${{\mathbb Y}}$ is non-singular if and only if $\Sigma_0 \cap \{\phi_d =0\} = \emptyset$. In this case the critical set of the map $\sigma : {{\mathbb Y}}\to {{\mathbb C}}$ is $\Sigma_0 \times {{\mathbb C}}\subset{{\mathbb Y}}_{0,{{\mathbb C}}}$.

2.  ${{\mathbb Y}}^{{\infty}}$ is non-singular if and only if $W_0 \cap \{\phi_d =0\} = \emptyset$. In this case the critical set of the map $\sigma_| : {{\mathbb Y}}^{{\infty}}\to {{\mathbb C}}$ is $W_0 \times {{\mathbb C}}\subset{{\mathbb Y}}^{{\infty}}_{0,{{\mathbb C}}}$.

\(a) We have that $(\Sigma_0 \cap \{ \phi_d = 0\}) \times {{\mathbb C}}= {{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}\cap \{ \sigma 
=0\}$. Since ${{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}$ is a germ of an analytic set at ${{\mathbb Y}}_{0,{{\mathbb C}}}$, it follows that ${{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}$ is empty if and only if $({{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}) \cap \{ \sigma =0\}$ is empty. The critical 