While reading 'Hyperelliptic Riemann surfaces' from 'Farkas and Kra', I came across a set of holomorphic functions on a compact Hyperelliptic Riemann surface $M$ of genus $g$ whose construction I've described below;
Say $z$ be a meromorphic function of degree $2$ on $M$. Say $P_1,P_1,....,P_{2g+2}$ ,be its $2g+2$ branch points ( these are exactly the Weierstrass points on $M$).We may assume that '$P_i$'s are neither zeroes and nor poles of $z$.Then the function $w=\sqrt{\Pi_{i=1}^{2g+2}(z-z(P_i))}$ can be proved to be single valued on $M$.
After this Farkas and Kra go on to claim that the $g$ functions $\frac{z^jdz}{w}, \ \ j\in \{0,1,...,g-1\}$ forms a basis for the space of holomorphic functions on $M$.
The proof as to why each of this is in fact an abelian differential of first kind is clear to me. But I'm not going anywhere on why these functions are linearly independent.
Say, $\Sigma_{i=0}^{g-1}c_i\frac{z^idz}{w}=0$. Then, $\Sigma_{i=0}^{g-1}c_iz^i=0$ everywhere. I feel like this polynomial being identically zero around a non-branch point of $z$ (where $z$ is thus the local coordinate) would mean that the coefficients would have to to be $0$ as well .
But ,the power $g-1$ has some important role in this entire business, as on inclusion of $\frac{z^gdz}{w}$ into the set above would make it linearly dependent. And then we'd have coefficients $c_1,c_2,...,c_g$ such that even at non-branch points $\Sigma_{i=0}^gc_iz^i=0$.
Can someone please enlighten me on this problem? Thanks in advance.