If $X$ is a smooth algebraic curve (over $\mathbb{C}$) with genus $g$, and $\omega\in \Omega^1(X)$ is a meromorphic 1-form, then it is a corollary of the Riemann-Roch theorem that $deg(div(\omega)) = 2g-2$.
My question is, if $\omega \in \Omega^{k}(X)$ is a higher order differential, what sort of estimates can we give to $deg(div(w))$ ?