Natural coordinate of a holomorphic differential
= Natural coordinate of a holomorphic differential
{title2=$dw^m=\phi(z)\,dz^m$}
= Flat coordinate
{synonym}
For a nonvanishing local $m$-differential $\phi(z)\,dz^m$, choose $w=\int\phi^{1/m}\,dz$. Then the differential is $dw^m$. Different root choices give $w\mapsto\zeta w+b$, $\zeta^m=1$. For $m=1,2$ the permitted linear parts are translations or signs, producing <translation surfaces> and <half-translation surfaces>.