locus of $\sigma : {{\mathbb Y}}\to {{\mathbb C}}$ is a germ at ${{\mathbb Y}}_{0,{{\mathbb C}}}$ and therefore it is ${{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}_{0,{{\mathbb C}}} = \Sigma_0 \times {{\mathbb C}}$. The proof of (b) is similar.

\[c:phi\] Let $f_0$ be a [[$\mathcal F$[[ISI]{}]{}]{}]{} polynomial and let $P$ be a deformation such that $W_0 \cap \{ \phi_d = 0\}$ is empty. Then ${{\mathbb Y}}_{s,t}$ and ${{\mathbb Y}}^{{\infty}}_{s,t}$ are non-singular, for all $t\in {{\mathbb C}}$ and all $s\not= 0$ close to $0$.

From Proposition \[l:phi\], it follows that $W_s = \emptyset$, for $s\not= 0$. By (\[eq:sing\]), we get that ${{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}_{s,{{\mathbb C}}}$ is empty and so is ${{\rm{Sing\hspace{2pt}}}}{{\mathbb Y}}^{{\infty}}_{s,{{\mathbb C}}}$. Since both spaces are products by the $t$-coordinate, we get our claim.

Since we shall use this several times, we should emphasize that in case $W_0 \cap \{ \phi_d = 0\}$ is empty, the spaces ${{\mathbb Y}}$ and ${{\mathbb Y}}^{{\infty}}$ are non-singular. This is in contrast to the compactifying spaces ${{\mathbb Y}}_{0,{{\mathbb C}}}$, resp. ${{\mathbb Y}}_{0,{{\mathbb C}}}^{{\infty}}$, of the original polynomial $f_0$, which spaces have non-isolated singularities as soon as $\Sigma_0 \not= \emptyset$, resp. $W_0 \not= \emptyset$. For instance, in ${{\mathbb Y}}$ we have smoothings ${{\mathbb Y}}_{s,{{\mathbb C}}}$ of ${{\mathbb Y}}_{0,{{\mathbb C}}}$ and in ${{\mathbb 