Let $f:\mathbb{C}\to\mathbb{C}$ be a complex polynomial $f=a_{0}+a_{1}z+...+a_{n}z^{n}$ without double zeroes. Consider for every natural number $k\geq 2$ the set $$M=\{(z,w)\in\mathbb{C}^{2}: w^{k}-f(z)=0\}$$ Prove that $M$ is a 2-dimensional submanifold of $\mathbb{C}^{2}\cong\mathbb{R}^{4}$.
A subset $M\subset\mathbb{R}^{n}$ is a $k$-dimensional submanifold if for every point $x\in M$ the following condition is satisfied:
There is an open set $U\subset\mathbb{R}^{n}$ containing $x$, an open set $V\subset\mathbb{R}^{n}$, and a diffeomorphism $h:U\to V$ such that $$h(U\cap M)=V\cap(\mathbb{R}^{k}\times \{0\})=\{y\in V:y^{k+1}=\cdots=y^{n}\}$$ So, how I can conclude that $M$ is a 2-dimensional submanifold..Thanks!
You can just show that the function $g(z, w)=w^k-f(z)$ is a submersion.