Consider a polynomial map $f:\mathbb{R}^{n-1}\to V\subset\mathbb{R}^n$ where $V$ is $n-1$-dimensional variety in $\mathbb{R}^n$.
Are there any conditions on $f$ to determine whether it defines bi-rational equivalence between $V$ and $\mathbb{R}^{n-1}$?
Thanks in advance!
P.S. I need to prove that some special class of polynomial maps are bi-rational. I am not getting any criterion to declare entire class of polynomial maps to be bi-rational. If somebody could point out to some references, that would also be helpful.