Elementary question in differential topology Let $p$ be a polynomial in $\mathbb{R}^n.$ Is the variety $Z(p)$ a $n-1$ dimensional manifold?another words why is that $o=(0,\cdots,0)$ a regular point for $p$ viewed as a smooth function from $\mathbb{R}^n$ to $\mathbb{R}$?

It follows from the implicit function theorem, that for a smooth $f:\mathbb{R}^m\to\mathbb{R}^n$ and a regular value $p\in\mathbb{R}^n$, the inverse image $f^{-1}(p)$ is a smooth manifold of dimension $m-n$. 
In general, if $p$ is not necessarily regular, $f^{-1}(p)$ needs not be a manifold.
