= Solution
Choose centred charts so the coordinate expression $F$ satisfies $F(0)=0$. On a component where the map is nonconstant, the <identity theorem> makes $F$ nonconstant. Its <Taylor expansion> factors as $F(z)=z^nh(z)$, with $n\ge1$ and $h(0)\ne0$. In a sufficiently small disc, choose a <holomorphic logarithm> of $h$, and hence a <holomorphic> $n$th root $q$ with $q^n=h$. The function $zq(z)$ has nonzero derivative at zero. The <holomorphic inverse function theorem>, which says that a <holomorphic function> with nonzero derivative has a <holomorphic> local inverse, makes it a valid new source coordinate. In that coordinate,
$$
\boxed{F(z)=z^n.}
$$
If the map is bijective, local injectivity forces $n=1$: for $n>1$, a small nonzero value has $n$ distinct preimages in this local disc. The inverse is therefore <holomorphic> in a neighbourhood of every point, and the local inverses agree, proving \b[the map is <biholomorphic>]. The nonconstant hypothesis must apply to the component containing $p$ if disconnected <Riemann surfaces> are allowed; otherwise a constant component is an exception to the proposed local form.
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