Proposition 4

Let $f: V\rightarrow W$ be a polynomial map. Assume that there exists $x_0\in V$ such that $(Df)(x_0)$ is surjective. Then $f$ is dominant.

Applying a translation in V and one in W, we can assume that $x_0= 0$ and $f(x_0) = 0$. The decomposition of $f$ as a sum of homogeneous elements can then be written

$f=f_1+f_2+\cdots$ with deg $f_i=i$,

and the linear map $f_1$ is surjective by hypothesis. Suppose that $f$ is not dominant. Then there exists a non-zero element $\psi$ of $A_W$ such that $\psi \circ f= 0$. Let $\psi =\psi_m+\psi_{m+1}+\cdots$ be the decomposition of $\psi$ into homogeneous elements, with deg $\psi_i=i$ and $\psi_m\not= 0$. Then

where $\rho$ is a sum of homogeneous polynomial maps of degrees $> m$. It follows that $\psi_m\circ f_1= 0$. Since $f_1$ is surjective, $\psi_m= 0$, a contradiction.