Let $X$ be the algebraic curve in $\mathbb{C}^{2}$ defined by the polynomial $p(z, w)=z^{d}+w^{d}+1$ where $d$ is a natural number. Using the implicit function theorem, or otherwise, show that there is a natural complex structure on $X$. Let $f: X \rightarrow \mathbb{C}$ be the function defined by $f(a, b)=b$. Show that $f$ is holomorphic. Find the ramification points and the corresponding branching orders of $f$.

Assume that $f$ extends to a holomorphic map $g: Y \rightarrow \mathbb{C} \cup\{\infty\}$ from a compact Riemann surface $Y$ to the Riemann sphere so that $g^{-1}(\infty)=Y \backslash X$ and that $g$ has no ramification points in $g^{-1}(\infty)$. State the Riemann-Hurwitz formula and apply it to $g$ to calculate the Euler characteristic and the genus of $Y$.