\ P(x,s) = t\}$, and get an algebraic set into ${{\mathbb P}}^n \times {{\mathbb C}}\times {{\mathbb P}}^1$, which we slice by $t={{\infty}}\in {{\mathbb P}}^1$ and by $\{ \sigma =0\}$. This gives the set $\{ \frac{\partial f_d}{\partial x} (x) = 0\}\subset H^{{\infty}}$.

For general-at-infinity deformations we can prove more precise results, as follows (see also Theorem \[t:lambdaexchange\]).

\[l:iomdin\] Let $P$ be a general-at-infinity deformation of a [[$\mathcal F$[[ISI]{}]{}]{}]{} $f_0$. If for some $t\in {{\mathbb C}}$, ${{\mathbb Y}}_{0,t}$ is non-singular but tangent to $H^{{\infty}}$ then $\Gamma_{III} \not= \emptyset$.

Our hypothesis on ${{\mathbb Y}}_{0,t}$ is equivalent to $\Sigma_0 =\emptyset$ and $W_0 \not= 
\emptyset$. Since $\Gamma_P \cap {{\mathbb Y}}_{0,t} \subset \Sigma_0 \times {{\mathbb C}}$, it follows that there are no type II components of $\Gamma_P$. Moreover, $\Sigma_0 =\emptyset$ implies that $\lambda(0) = 0$. Since $W_0 \not= \emptyset$, the total Milnor number $\mu(0)$ is strictly less than $\mu(s)$, for $s\not= 0$, by (\[eq:chi\]) and (\[eq:gammas\]). This can only be due to the fact that affine critical values of $f_s$ tend to infinity, as $s\to 0$. This means that branches of type III do exist.

\[l:tang\] Let $P$ be a one-parametre [[$\mathcal F$[[ISI]{}]{}]{}]{} deformation and assume that the total space ${{\mathbb Y}}$ is non-singular. If for 