Let $n \in \mathbb{N}$, $f\in \mathbb{C}[X_1,\dots,X_n]$, and $D(f):=\{x=(x_1,\dots,x_n)\in \mathbb{C^n}|f(x)\neq 0\}.$ I want to show that there is a injective $\Phi: D(f) \rightarrow \mathbb{C}^{n+1}$ so that there are $g_1,\dots g_k \in \mathbb{C}[X_1,\dots,X_{n+1}]$ with $\Phi(D(f))=\{x=(x_1,\dots,x_{n+1})\in \mathbb{C^n+1}| g_i(x)=0 \forall i=1,\dots,k\}$.
I think that $\Phi$ should be defined as $\Phi(x_1,\dots x_n)=(x_1,\dots,x_n,f((x_1,\dots,x_n))$ and that $g_1(x_1, \dots, x_{n+1})=f(x_1,\dots,x_n)-x_{n+1}$ makes sense. But I it doesn't work if I pick $(x_1,\dots,x_n,0)$ with $f(x_1,\dots, x_n)=0$. So, am I on the right track in if so, what would make a good $g_2, g_3 \dots$.
Thank you
It would be like $\Phi(x_1,\dots x_n)=(x_1,\dots,x_n,1/f(x_1,\dots,x_n))$, namely $x_{n+1}*f(x_1,\dots,x_n)=1$