if and only if ${{\mathbb Y}}_{s,t}$ and ${{\mathbb Y}}^{{\infty}}_{s,t}$ are non-singular, for all $t\in {{\mathbb C}}$ and all $s\not= 0$ close to $0$. Then our claim follows from Corollary \[c:phi\].

The affine critical locus
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In case of several parameters, we have of course more room for constructing generic deformations. For instance, let $P\colon {{\mathbb C}}^n\times {{\mathbb C}}^n \to {{\mathbb C}}$ be a deformation of $f$ such that $P_d= f_d + 
s_1x_1^d + \cdots + s_nx_n^d$. Then ${{\mathbb Y}}$ is nonsingular and ${{\mathbb Y}}_{{{\mathbb C}},t}$ is nonsingular too, for any $t$.

Let us now focus on the critical locus of the deformation.

\[d:gamma\] Let $\Psi := (\sigma, \tau) :  {{\mathbb Y}}\setminus {{\mathbb Y}}^{{\infty}}\to {{\mathbb C}}^k \times {{\mathbb C}}$ be the map induced by the couple of projections. We define the [*affine critical locus*]{} $\Gamma_P$ to be the closure in ${{\mathbb P}}^n \times {{\mathbb C}}^k \times {{\mathbb P}}$ of the set of points where $\Psi$ is not a submersion. Let us call [*affine discriminant*]{} and denote by $\Delta_P$ the closure in ${{\mathbb C}}^k \times {{\mathbb P}}$ of the image $\Psi( \Gamma_P \cap ({{\mathbb C}}^n \times {{\mathbb C}}^k \times {{\mathbb C}}))$ of the affine part of $\Gamma_P$.

Since we are only interested 