Let \(f:\mathbb C\to\mathbb C\) be a polynomial of degree at least \(2\). Let \[ V_1=\{f(z):f'(z)=0\},\qquad V_0=f^{-1}(V_1),\qquad U_i=\mathbb C\setminus V_i\quad(i=0,1). \] Prove that \(f:U_0\to U_1\) is a covering map.
Solution
1. The sets \(V_1\) and \(V_0\) are finite, and\[ U_0=f^{-1}(U_1). \]
If \(d=\deg f\ge2\), then \(f'\) is a nonzero polynomial of degree \(d-1\), so it has only finitely many zeros. Therefore its image \[ V_1=\{f(z):f'(z)=0\} \] is finite. For each \(w\in V_1\), the equation \(f(z)=w\) has at most \(d\) solutions, so \[ V_0=f^{-1}(V_1) \] is finite as well. Finally, \[ f^{-1}(U_1) =f^{-1}(\mathbb C\setminus V_1) =\mathbb C\setminus f^{-1}(V_1) =U_0. \]
2. The polynomial map \(f:\mathbb C\to\mathbb C\) is proper.
Write \[ f(z)=a_dz^d+a_{d-1}z^{d-1}+\cdots+a_0, \qquad a_d\ne0. \] Then \[ \frac{f(z)}{z^d} =a_d+\frac{a_{d-1}}z+\cdots+\frac{a_0}{z^d} \longrightarrow a_d \] as \(|z|\to\infty\). Hence \(|f(z)|\to\infty\) as \(|z|\to\infty\).
If \(K\subset\mathbb C\) is compact, it is bounded, so the preceding limit shows that \(f^{-1}(K)\) is bounded. It is also closed because \(f\) is continuous and \(K\) is closed. Thus \(f^{-1}(K)\) is compact in \(\mathbb C\).
3. The restriction\[ f:U_0\longrightarrow U_1 \]is a proper local homeomorphism.
2. Hence the restriction is proper.
If \(z\in U_0\), then \(f'(z)\ne0\): otherwise \(f(z)\in V_1\), which would put \(z\) in \(V_0\). The complex inverse function theorem therefore gives a neighborhood of every \(z\in U_0\) on which \(f\) is a homeomorphism onto an open neighborhood of \(f(z)\). Thus the restriction is a local homeomorphism.
4. The target \(U_1\) is connected and the restriction is a covering map.
By 1, \(U_1\) is the complex plane with finitely many points removed. Any two points of such a set can be joined by a polygonal path avoiding those finitely many points, so \(U_1\) is path-connected and hence connected. Likewise \(U_0\) is nonempty because it is the complement of the finite set \(V_0\).
By 3, \(f:U_0\to U_1\) is a proper local homeomorphism. By E-HM213A17-04, a proper local homeomorphism between open subsets of \(\mathbb C\) with nonempty domain and connected target is surjective and is a covering map. Therefore \[ f:U_0\longrightarrow U_1 \] is a covering map.