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 $\mathbb{R}$?

In general, no. $Z(x^2+y^2)= \{(0,0)\}$ is not a $1$-dimensional manifold.
