# Local degree of a polynomial defined on the Riemann sphere at a root

I'm working on a problem from Hatcher's Algebraic Topology, and I want to show that if we take polynomial $f(x)$ definded on the Riemann sphere mapping to the Riemann sphere, the local degree of the map (when viewed as a map $S^2 \to S^2$) at a root is equal to the multiplicity of the root.

Intuitively this seems obvious because if we look at a neighborhood of some root with multiplicity $n$ and translate the function to the origin, we get a map that kind of looks like the map $z \to z^n$, given a sufficiently small neighborhood, which has degree $n$ as desired. My problem is making my claim rigorous because the polynomial does not map perfectly from $z \to z^n$ in the neighborhood. I'm not sure how to compute the local degree directly. I know it means looking at the induced map $H_2(U,U-x_0) \to H_2(f(U), f(U)-0)$ where $x_0$ is a root and $U$ is a neighborhood of $x_0$.

We can use the argument principle, which states the number of zeros minus the number of poles around a small contour around the root $x$ is the change in the argument of $f(z)$ as $z$ travels around the contour, divided by $2 \pi$. This means the multiplicity of $x$ is number of times $f(z)$ travels around $0$ when $z$ travels around a sufficiently small contour surrounding $x$. In other words, this means the local degree of $f$ at the root is the multiplicity of the root as desired.
Here's a way to do this without complex analysis, which is based on the idea that $$f$$ "looks like" (i.e., is homotopic in some sense to) the map $$z \mapsto z^n$$ near the root.
If $$f$$ has a root at $$a \in \mathbb{C}$$, we can write $$f(z) = \lambda (z - a)^k(z - b_1) \cdots (z - b_m)$$ by the fundamental theorem of algebra. For simplicity, let's assume $$a = 0$$ and $$\lambda = 1$$. We have a homotopy $$h_t(z) = z^k(tz - b_1) \cdots (tz - b_m)$$ from the map $$g(z) = \beta z^k$$ (where $$\beta = b_1 \cdots b_m$$) to $$f$$. Observe that if $$z$$ is close to $$0$$, each term $$tz - b_j$$ is close to $$b_j$$, and in particular is nonzero. Therefore, for a sufficiently small neighborhood $$U$$ of $$0$$, the homotopy $$h_t$$ restricts to a homotopy $$h_t : (U, U - 0) \to (\mathbb{C}, \mathbb{C} - 0)$$. It follows that the induced map $$f_* : H_*(U, U - 0) \to (\mathbb{C}, \mathbb{C} - 0)$$ is the same as the induced map $$g_* : H_*(U, U - 0) \to (\mathbb{C}, \mathbb{C} - 0)$$. Note that these are the maps involved in the computation of the local degrees of $$f$$ and $$g$$, respectively. Thus, the local degree of $$f$$ at $$0$$ is the same as the local degree of $$g$$ at $$0$$, which we know is $$k$$.
Note that it is crucial that the maps $$h_t$$ send $$U - 0$$ to $$\mathbb{C} - 0$$, since that allows us to invoke the homotopy invariance of singular homology. Thus, we are using a fact that is stronger than $$f$$ and $$g$$ merely being homotopic (which is rather trivial since $$\mathbb{C}$$ is contractible).