(a) Let $f: R \rightarrow S$ be a non-constant holomorphic map between compact connected Riemann surfaces $R$ and $S$.

Define the branching order $v_{f}(p)$ at a point $p \in R$ and show that it is well-defined. Show further that if $h$ is a holomorphic map on $S$ then $v_{h \circ f}(p)=v_{h}(f(p)) v_{f}(p)$.

Define the degree of $f$ and state the Riemann-Hurwitz formula. Show that if $R$ has Euler characteristic 0 then either $S$ is the 2 -sphere or $v_{p}(f)=1$ for all $p \in R$.

(b) Let $P$ and $Q$ be complex polynomials of degree $m \geq 2$ with no common roots. Explain briefly how the rational function $P(z) / Q(z)$ induces a holomorphic map $F$ from the 2-sphere $S^{2} \cong \mathbb{C} \cup\{\infty\}$ to itself. What is the degree of $F$ ? Show that there is at least one and at most $2 m-2$ points $w \in S^{2}$ such that the number of distinct solutions $z \in S^{2}$ of the equation $F(z)=w$ is strictly less than $\operatorname{deg} F$.