On a singular variety associated to a polynomial mapping from ${\text{C}}^n$ to ${\text{C}}^{n-1}$

We construct a singular variety ${\mathcal{V}}_G$ associated to a polynomial mapping $G:{\text{C}}^n\to {\text{C}}^{n-1}$ where $n\ge 2$. We prove that in the case $G:{\text{C}}^3\to {\text{C}}^2$, if $G$ is a local submersion but is not a fibration, then the homology and the intersection homology with total perversity (with compact supports or closed supports) in dimension two of the variety ${\mathcal{V}}_G$ is not trivial. In the case of a local submersion $G : \C^{n} \to \C^{n - 1}$where$n \geq 4$, the result is still true with an additional condition.