Are the fibers of a surjective polynomial submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic? Are the fibers of a surjective polynomial submersion $\mathbb{C}^n\to\mathbb{C}$ all homeomorphic?
 A: The following example is taken from Section 7 of Nguyen, "A Remark on Polynomial Mappings from $\mathbb{C}^n$ to $\mathbb{C}^{n-1}$ and an Application of the Software Maple in Research": Map $\mathbb{C}^2$ to $\mathbb{C}$ by $f(z,w) = z+z^2w$.
We compute $\tfrac{\partial f}{\partial z} = 1+2zw$ and $\tfrac{\partial f}{\partial w} = z^2$; these two polynomials have no common zero, so $f$ is a submersion.
We have $f^{-1}(0) = \{ z(1+zw) = 0 \}$, which is the disjoint union of the once-punctured genus zero curve $z=0$ and the twice-punctured genus zero curve $zw=-1$.
For $\lambda \neq 0$, the equation $z^2 w + z = \lambda$ is a twice punctured genus zero curve. It can be parameterized as $z \mapsto (z, (\lambda-z)/z^2)$ for $z \neq 0$.
It looks like Nguyen's paper has other interesting examples as well.

Other interesting references:
Artal-Bartolo, Cassou-Nogues and Luengo-Velasco, On polynomials whose fibers are irreducible with no critical points, construct submersions $\mathbb{C}^2 \to \mathbb{C}$  whose generic fibers have arbitrarily large genus. I suspect that not all the fibers are homeomorphic, but don't understand their construction, so I don't know.
If I understand the Mathscinet review of "Polynomials with general $C^2$-fibers are variables" correctly, then Abyhankar and Moh "Embeddings of the line in the plane" and Suzuki "Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace $C^2$" show that, if $f(x,y)$ is an irreducible polynomial such that $\{ f(x,y)  = 0 \}$ is isomorphic to $\mathbb{C}$, then there is an automorphism of $\mathbb{C}^2$ taking $f$ to one of the coordinate functions; this would then imply that $\{ f(x,y) = \lambda \}$ is isomorphic to $\mathbb{C}$ for all $\lambda$. I have not attempted to read the papers in question.
