If $F\in\mathbb{C}[x_{1},\ldots,x_{n+1}]$ is a smooth homogeneous polynomial, is it true that the dehomogenisation $F(x_{1},\ldots ,x_{n},1)\in\mathbb{C}[x_{1},\ldots,x_{n}]$ is also smooth?
Smooth in this context meaning non-singular.
I know that the converse is false; i.e. if we consider a smooth polynomial $f$ then its homogenisation is not always smooth.