Proposition 3

Assume that $k$ is algebraically closed. Let $f: V\rightarrow W$ be a dominant polynomial map. The image under $f$ of any dense open subset of V contains a dense open subset of W.

It suffices to prove that, for every non-zero element $\varphi$ of $A_V,f(V_{\varphi})$ contains a dense open subset of W. Identify $A_W$ with a subalgebra of $A_V$ by means of the homomorphism associated to $f$. There exists a non-zero element $\psi$ of $A_W$ such that every homomorphism $w: A_W\rightarrow k$ which does not annihilate $\psi$ extends to a homomorphism $v: A_V\rightarrow k$ which does not annihilate $\varphi ($Commutative Algebra, Chap. V, §3, no. 1, Cor. 3 of Th. 1). Now such a $w$ (resp. $v$) can be identified with an element of $W_{\psi}$ (resp. of $V_{\varphi}$) and to say that $v$ extends $w$ means that $f(v) =w$. Hence, $W_{\psi}\subset f(V_{\varphi})$. Q.E.D.

Let $f: V\rightarrow W$ be a polynomial map, and $x_0\in V$. The map $h \rightarrow f(x_0+h)$ from V to W is polynomial. Decompose it into a finite sum of homogeneous polynomial maps:

where $D_i: V\rightarrow W$ is homogeneous of degree $i($Algebra, Chap. IV, §5, no. 10, Prop. 19). The linear map $D_1$ is called the tangent linear map of $f$ at $x_0$. We denote it by $Df(x_0)$.