Question: If I have a non-zero polynomial $P(x_1,\ldots,x_n)$ defined on $\mathbb R^n$, is it true that the zero set $$Z(P) = \{x \in \mathbb R^n: P(x_1,\ldots,x_n) = 0\}$$ can be written as finite union of smooth submanifolds of $\mathbb R^n$?
This is probably a very basic thing of real algebraic geometry, but I'm totally ignorant in the area. If the response is affirmative, can anyone provide a reference?