I want to investigate the ring $R$ of holomorphic functions on a Riemann surface $X$ from an algebraic point of view. A consideration on the valuations corresponding to the localization after maximal ideals lead me to the following complex analytic question:
Question: Are there examples of Riemann surfaces X such that we can prescribe positive integers and points in $X$ such that some holomorphic function has orders (degrees) equal to the prescribed numbers in the given points? If yes, how large can the set of all prescribed points be?
For instance: Given distinct points $p_1,...,p_n \in X$ and $n_1,...,n_m \in \mathbb{N}$ can we find some $f \in R$ such that $ord_{p_i}(f) = n_i$ for all $ i \in {1,...,m}$?
Or: Can we do this even for infinite families of points etc. ?
I am not so familiar with complex analysis and have therefore no chance to answer this question on my own. Thanks in advance for any help!