some $t\in {{\mathbb C}}$, ${{\mathbb Y}}_{0,t}$ has singularities on $H^{{\infty}}$ then all branches of $\Delta_P$ of type II are tangent to the $\{\sigma = 0\}$-axis.

Our condition on ${{\mathbb Y}}_{0,t}$ is equivalent to $\Sigma_0 \not=\emptyset$ and is a necessary condition for the existence of type II branches of $\Gamma_P$, see Lemma \[l:curve2\]. Let $v= (p,t_0)\in \Gamma_P \cap {{\mathbb Y}}^{{\infty}}$. Denote by $\Gamma_{P,v}$ the germ of $\Gamma_P$ at $v$. We may assume that $\Gamma_{P,v}$ is an irreducible germ, if not, then we take an irreducible component of it. Take a parametrization of $\Gamma_{P,v}$, $\gamma({\varepsilon})$, with $\gamma(0)=v$. One should prove that the fraction $\sigma (\gamma({\varepsilon}))/ \tau(\gamma({\varepsilon}))$ tends to 0 as ${\varepsilon}\to 0$. This is indeed so since $\tau$ is a linear function germ of the non-singular space ${{\mathbb Y}}$ and $\sigma$ has a singularity at $v$, hence it is in the square of the maximal ideal of holomorphic germ at $v$. Then our claim follows by Lê’s result [@Le-c Proposition 1.2].

Changes of topology within deformations
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We study the change of topology within the class of [[$\mathcal F$[[ISI]{}]{}]{}]{} polynomials. Let $P$ be a [[$\mathcal F$[[ISI]{}]{}]{}]{} deformation of a polynomial $f=f_0$ of degree $d$. We recall the notations in §\[s:inv\] and adapt 