Let $X$ be a compact Riemann surface of genus $g$. We identify it with $4g$ polygon $\{a_i,b_i, a_i^\prime, b_i^\prime\}_{i=1}^g$. For a meromorphic 1 form $\omega$, we define $A_i(\omega)= \int_{a_i}\omega$, and $B_i(\omega)= \int_{b_i}\omega$. $A_i$ are called $a$ -periods and $B_i$ are called $b$-periods.

Suppose we take finitely many points $p_i$ and a finite set of Laurent tails $r_i(z)$ having only negative terms; we know that there is a meromorphic 1-form $\omega$ on $X$ whose Laurent series has $r_i(z) dz_i$ as its terms of negative degree for each $i$ if and only if the sum of the coefficients of $z_i^{-1}dz_i$ terms are zero.

Question: $p_i$ finitely many points are fixed, Laurent tails as in the previous paragraph are fixed, and $\omega$ one form as in the previous paragraph; now what extra condition in terms of $a-periods$ or $b-Periods$, we need to give so that $\omega$ (as in the pervious para) is unique.

(Attempt: I am trying to modify the proof of Lemman 4.6, page 260, Rick Miranda-- and looking for the suggestion- in case it is trivial)

Let $X$ be a compact Riemann surface of genus $g$. We identify it with $4g$ polygon $\{a_i,b_i, a_i^\prime, b_i^\prime\}_{i=1}^g$. For a meromorphic 1 form $\omega$, we define $A_i(\omega)= \int_{a_i}\omega$, and $B_i(\omega)= \int_{b_i}\omega$. $A_i$ are called $a$ -periods and $B_i$ are called $b$-periods.

Suppose we take finitely many points $p_i$ and a finite set of Laurent tails $r_i(z)$ having only negative terms; we know that there is a meromorphic 1-form $\omega$ on $X$ whose Laurent series has $r_i(z) dz_i$ as its terms of negative degree for each $i$ if and only if the sum of the coefficients of $z_i^{-1}dz_i$ terms are zero.

Question: $p_i$ finitely many points are fixed, Laurent tails as in the previous paragraph are fixed, and $\omega$ one form as in the previous paragraph; now what extra condition in terms of $a$-periods or $b$-periods, we need to give so that $\omega$ (as in the pervious para) is unique.

(Attempt: I am trying to modify the proof of Lemma 4.6, page 260, Rick Miranda-- and looking for the suggestion- in case it is trivial)

Let $X$ be a compact Riemann surface of genus $g$. We identify it with $4g$ polygon $\{a_i,b_i, a_i^\prime, b_i^\prime\}_{i=1}^g$. For a meromorphic 1 form $\omega$, we define $A_i(\omega)= \int_{a_i}\omega$, and $B_i(\omega)= \int_{b_i}\omega$. $A_i$ are called $a$ -periods and $B_i$ are called $b$-periods.

Suppose we take finitely many points $p_i$ and a finite set of Laurent tails $r_i(z)$ having only negative terms; we know that there is a meromorphic 1-form $\omega$ on $X$ whose Laurent series has $r_i(z) dz_i$ as its terms of negative degree for each $i$ if and only if the sum of the coefficients of $z_i^{-1}dz_i$ terms are zero.

Question: $p_i$ finitely many points are fixed, Laurent tails as in the previous paragraph are fixed, and $\omega$ one form as in the previous paragraph; now what extra condition in terms of $a-periods$ or $b-Periods$, we need to say that such (as in the pervious para) $\omega$ is unique.

(Attempt: I am trying to modify the proof of Lemman 4.6, page 260, Rick Miranda-- and looking for the suggestion- in case it is trivial)

Let $X$ be a compact Riemann surface of genus $g$. We identify it with $4g$ polygon $\{a_i,b_i, a_i^\prime, b_i^\prime\}_{i=1}^g$. For a meromorphic 1 form $\omega$, we define $A_i(\omega)= \int_{a_i}\omega$, and $B_i(\omega)= \int_{b_i}\omega$. $A_i$ are called $a$ -periods and $B_i$ are called $b$-periods.

Suppose we take finitely many points $p_i$ and a finite set of Laurent tails $r_i(z)$ having only negative terms; we know that there is a meromorphic 1-form $\omega$ on $X$ whose Laurent series has $r_i(z) dz_i$ as its terms of negative degree for each $i$ if and only if the sum of the coefficients of $z_i^{-1}dz_i$ terms are zero.

Question: $p_i$ finitely many points are fixed, Laurent tails as in the previous paragraph are fixed, and $\omega$ one form as in the previous paragraph; now what extra condition in terms of $a-periods$ or $b-Periods$, we need to give so that $\omega$ (as in the pervious para) is unique.

(Attempt: I am trying to modify the proof of Lemman 4.6, page 260, Rick Miranda-- and looking for the suggestion- in case it is trivial)