Y}}_{{{\mathbb C}}, t}$ we have smoothings ${{\mathbb Y}}_{s,t}$ of ${{\mathbb Y}}_{0,t}$, for all $t$, for $s\not=0$.

\[ex:yomdin\] [*Yomdin deformations*]{} of $f_0$ are of the form: $$\ P = f_0  - sx_1^d,$$ where $x_1$ denotes a coordinate. Sufficiently general Yomdin deformations, i.e. such that $\Sigma_0 \cap \{ x_1 = 0\} = \emptyset$, are [[$\mathcal F$[[ISI]{}]{}]{}]{} deformations. We can use the chart $x_1 =1$ to obtain a description of ${{\mathbb Y}}$ as graph of a function: $$s = (f_0)_d(1, x_2, \ldots, x_n) + x_0(f_0)_{<d} - t x_0^d,$$ where $(f_0)_{<d}$ denotes $(f_0)_{d-1} + x_0(f_0)_{d-2}+\cdots$.

Generic deformations
--------------------

\[d:gi\] A polynomial $f : {{\mathbb C}}^n \to {{\mathbb C}}$ such that, for any $t\in {{\mathbb C}}$, the compactified fiber $\overline{f^{-1}(t)} \subset {{\mathbb P}}^n$ is transversal to the hyperplane at infinity $H^{{\infty}}= 
{{\mathbb P}}^n\setminus {{\mathbb C}}^n$, will be called [*general-at-infinity*]{} (abbreviated GI). We say that a 1-parameter deformation $P:{{\mathbb C}}^n \times {{\mathbb C}}\to {{\mathbb C}}$ is GI if the polynomial $f_s$ is GI, for any $s\not= 0$ within a small neighbourhood of $0$.

We have the following property.

\[p:max\] The general fiber $G$ of a general-at-infinity polynomial $f$ is homotopy equivalent to a bouquet of $(n-1)$-spheres and the number of spheres is $\mu = 
b_{n-1}(G) = 