A polynomial \(f(z)\) with complex coefficients, viewed as a map \(\mathbb{C} \to \mathbb{C}\), can always be extended to a continuous map of one-point compactifications \(\hat{f}: S^2 \to S^2\). Show that the degree of \(\hat{f}\) equals the degree of \(f\) as a polynomial. Show also that the local degree of \(\hat{f}\) at a root of \(f\) is the multiplicity of the root.
Solution
Let \[ f(z)=a_mz^m+a_{m-1}z^{m-1}+\cdots+a_0, \qquad a_m\ne0. \]
1. The map \(f:\mathbb C\to\mathbb C\) is proper and therefore extends continuously to one-point compactifications by setting\[ \widehat f(\infty)=\infty. \]
As \(|z|\to\infty\), \[ |f(z)|\ge |a_m||z|^m-\sum_{j<m}|a_j||z|^j\longrightarrow\infty. \] Hence inverse images of compact sets are bounded and closed, therefore compact. This is exactly the condition for extension to the one-point compactifications.
2. The map \(\widehat f\) is homotopic to the compactification of the leading monomial \(a_mz^m\).
Consider \[ f_t(z)=a_mz^m+t(a_{m-1}z^{m-1}+\cdots+a_0), \qquad0\le t\le1. \] The leading coefficient is independent of \(t\), so the estimate in 1 is uniform in \(t\) outside a sufficiently large disk. Thus the \(f_t\) extend to a continuous based homotopy of one-point compactifications.
3. The compactified map \(z\mapsto a_mz^m\) has degree \(m\).
Multiplication by \(a_m\ne0\) is orientation preserving as a real linear map of \(\mathbb C\), so it has degree \(+1\). For a nonzero regular value \(w\), the equation \[ z^m=w \] has exactly \(m\) distinct roots. Near each root the derivative \(mz^{m-1}\) is nonzero and complex multiplication by a nonzero complex number preserves the real orientation. Hence each preimage contributes local degree \(+1\), so the total degree is \(m\).
Therefore \[ \boxed{\deg\widehat f=m=\deg(f)\text{ as a polynomial}.} \]
4. If \(z_0\) is a root of multiplicity \(k\), then the local degree of \(\widehat f\) at \(z_0\) is \(k\).
Factor \[ f(z)=(z-z_0)^k g(z), \qquad g(z_0)\ne0. \] Choose a small circle \(|z-z_0|=\varepsilon\) on which \(g\) has no zero. The local degree equals the winding number around \(0\) of the image of this circle under \(f\). The factor \((z-z_0)^k\) winds exactly \(k\) times. The loop \(g(z)\) is homotopic in \(\mathbb C^*\) to the constant loop \(g(z_0)\) when \(\varepsilon\) is small, so it contributes winding number \(0\). Hence the total winding number, and therefore the local degree, is \(k\).