# Application of winding number and the roots of complex polynomial from a non simple closed cuvre

There is a formula for the simple closed curve $$\gamma(t)$$ and complex polynomial $$p(z)$$. The winding number of $$p(\gamma)$$ around (0,0) is the sum of roots counting multiplicity of $$p(z)$$ within the closed curve $$\gamma(t)$$. This is just a direct result from the argument principle.

Now I want to ask what if the closed curve $$\gamma(t)$$ is NOT simple anymore which means it may have self-intersections. What will the winding number $$w(p(\gamma),(0,0))$$ be? I think it should be still related to the roots of $$p(z)$$

The winding number of $$p \circ \gamma$$ with respect to $$0$$ is $$I(p \circ \gamma,0) = \frac{1}{2\pi i}\int_{p \circ \gamma} \frac{dw}{w} = \frac{1}{2\pi i}\int_\gamma \frac{p'(z)}{p(z)} \, dz$$ and that can be computed with the Residue theorem. If $$a_1, \ldots, a_m$$ are the distinct roots of $$p$$ with multiplicities $$k_1, \ldots ,k_m$$, then $$\frac{1}{2\pi i}\int_\gamma \frac{p'(z)}{p(z)} \, dz = \sum_{j=1}^m I(\gamma, a_j) \operatorname{Res}(\frac{p'}{p}, a_j) \, .$$ It is relatively easy to compute that the residue of $$p'/p$$ at a $$k$$-fold zero of $$p$$ is $$k$$, so that $$I(p \circ \gamma,0)= \sum_{j=1}^m I(\gamma, a_j) k_j \, .$$ This is the general formula.
For a simple positively oriented closed curve $$\gamma$$ the winding numbers $$I(\gamma, a_j)$$ are either zero or one, and the sum reduces to $$I(p \circ \gamma,0) = \sum_{a_j \text{ inside } \gamma} k_j$$