Given a irreducible polynomial $f_{n}+f_{n-1}$, where $f_{n}$ and $f_{n-1}$ are homogeneous polynomial of degree $n$ and $n-1$ , respectively. Then $f_{n}+f_{n-1}$ define a hypersurface $\mathit{S}\,$ in $\mathbb{A}^n$. Show that $\mathit{S}\,$ is birational to $\mathbb{A}^{n-1}$
I want to show that the map $$F:\mathbb{A}^{n-1} \to\mathit{S}\,$$ defined by $$(a_{1},a_{2},...,a_{n-1})\mapsto(\frac{-f_{n-1}}{f_{n}}(a_{1},a_{2},...,a_{n-1},1)a_{1},\frac{-f_{n-1}}{f_{n}}(a_{1},a_{2},...,a_{n-1},1)a_{2},...,\frac{-f_{n-1}}{f_{n}}(a_{1},a_{2},...,a_{n-1},1))$$
is the required map. But i had a problem to show that the image of $\mathbb{A}^{n-1}$ is dense in $\mathit{S}\,$.
That is, to show that the induced map from $k[\mathit{S}]\,$to $k[\mathbb{A}^{n-1}]=k[a_{1},a_{2},...,a_{n-1}]\,$is injective . But how to show that there is no non-trivial polynomial $p$ not in $<f_{n}+f_{n-1}>$ such that $p((\frac{-f_{n-1}}{f_{n}}(a_{1},a_{2},...,a_{n-1},1)a_{1},\frac{-f_{n-1}}{f_{n}}(a_{1},a_{2},...,a_{n-1},1)a_{2},...,\frac{-f_{n-1}}{f_{n}}(a_{1},a_{2},...,a_{n-1},1))=0$ ?
The surface defined above is named monoid surface.
Consider the projection from the origin(that is,solving the equation $t^{n}f_{n}+t^{n-1}f_{n-1}=0$ )map $F$ map $\mathbb{A}^{n-1}$ onto the complement of $V(f_{n-1})$ (or a smaller nonempty open set contain in it) in $S$ , that is , the image of $\mathbb{A}^{n-1} $ is dense in $S$.