(a) Let $X=\mathbb{C} \cup\{\infty\}$ be the Riemann sphere. Define the notion of a rational function $r$ and describe the function $f: X \rightarrow X$ determined by $r$. Assuming that $f$ is holomorphic and non-constant, define the degree of $r$ as a rational function and the degree of $f$ as a holomorphic map, and prove that the two degrees coincide. [You are not required to prove that the degree of $f$ is well-defined.]

Let $A=\left\{a_{1}, a_{2}, a_{3}\right\}$ and $B=\left\{b_{1}, b_{2}, b_{3}\right\}$ be two subsets of $X$ each containing three distinct elements. Prove that $X \backslash A$ is biholomorphic to $X \backslash B$.

(b) Let $Z \subset \mathbb{C}^{2}$ be the algebraic curve defined by the vanishing of the polynomial $p(z, w)=w^{2}-z^{3}+z^{2}+z$. Prove that $Z$ is smooth at every point. State the implicit function theorem and define a complex structure on $Z$, so that the maps $g, h: Z \rightarrow \mathbb{C}$ given by $g(z, w)=w, h(z, w)=z$ are holomorphic.

Define what is meant by a ramification point of a holomorphic map between Riemann surfaces. Give an example of a ramification point of $g$ and calculate the branching order of $g$ at that point.