Computer Algebra
《Computer Algebra》 · 第 72 页 · Chapter III，Section 18「The Abel–Jacobi map」，Theorem 18.4
Let $P_1,\ldots,P_r$ and $Q_1,\ldots,Q_r$ be points on $C$ (not necessarily distinct). Then there exists a meromorphic function $f$ on $C$ with poles exactly at the $P_i$ and zeros exactly at the $Q_i$ if and only if, for all paths $\gamma_i$ from $P$ to $P_i$ and all paths $\gamma_i'$ from $P$ to $Q_i$, there exists an element $\gamma\in H_1(C,\mathbb Z)$ such that $\sum_i\int_{\gamma_i}\omega-\sum_i\int_{\gamma_i'}\omega=\int_\gamma\omega$ for all $\omega\in\Gamma(C,\Omega^1)$.