Is it true that a polynomial submersion $ p: \mathbb{R}^2 \to \mathbb{R}$ of degree $n$ has at most $n$ connected components on each level?
I think I have a proof, can someone point me out any mistakes?
Let $c\in \mathbb{R}$.
Since $p$ is a submersion, $p-c$ doesn't have a $(x^2+y^2-R)$ factor for any $R\in \mathbb{R}^+$. Also, each connected component of $p^{-1}(c)$ must intersect $x^2+y^2-R=0$ at least twice, for every $R$ big enough.
By Bezout theorem, the system \begin{cases} p=c \\ x^2+y^2=R\end{cases}
has at most $2n$ real solutions, hence $p^{-1}(c)$ has at most $n$ connected components.