F$[[ISI]{}]{}]{}]{} polynomials
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In the following we consider a constant degree deformation of a polynomial $f_0$, i.e. a family $P:{{\mathbb C}}^n \times {{\mathbb C}}^k \to {{\mathbb C}}$ of polynomial functions $P(x,s) = 
f_s(x)$ such that $\deg f_s =d$ for all values of the parameter $s$ in a small neighbourhood of $0\in{{\mathbb C}}^k$. We assume that $P$ is holomorphic or polynomial in the parameters $s$ and polynomial in the variables $x$.

In this section we work with one-parameter deformations, i.e. $k=1$. The following sets play an important role in our study:

$$\label{eq:not}
\begin{array}{c}  W_s := \{ [x] \in {{\mathbb P}}^{n-1} \mid   \frac{\partial P_d}{\partial 
x}(x,s)=0\},\\ \\
 \Sigma_s := \{ [x] \in {{\mathbb P}}^{n-1} \mid   P_{d-1}(x,s)=0 \} \cap W_s,
 \end{array}$$

where $P_q$ stays for the degree $q$ homogeneous part of $P$ in variables $x\in {{\mathbb C}}^n$.

That $f_0$ is a [[$\mathcal F$[[ISI]{}]{}]{}]{} polynomial is equivalent to: $\dim W_0 \le 0$ and $\dim {{\rm{Sing\hspace{2pt}}}}f_0 
\le 0$. Remark that the second condition is not a consequence of the first one (example: $f_0 
= x^2$, as polynomial of $2$ variables).

\[d:main\] We say that $P$ is a [*[[$\mathcal F$[[ISI]{}]{}]{}]{} deformation*]{} of $f_0$ if $P$ has the property that $\dim W_0 \le 0$ and that $\dim {{\rm{Sing\hspace{2pt}}}}f_s 