The following question arises when I am trying to construct certain meromorphic modular functions on some congruence subgroup.
Question: Suppose one has a compact Riemann surface $X$ with genus $g$ and two meromorphic functions $f_1$,$f_2$ on it with prescribed poles at $p_1$ and $q_1,q_2,\cdots,q_{g+1}$. Is there an algorithm to construct a rational function of $f_1,f_2$ on $X$ with only one pole of given order $m$ at $p_1$?($m$ is different from the order $k$ of the only pole of $f_1$)