Computing a basis for $\mathcal{L}(G)$

Let $G$ be any rational divisor on ${\widetilde{C}}$. The algorithms of the preceding sections finally allow to compute an $\mathbf{F}$-basis of $\mathcal{L}(G)$ by the Brill-Noether Algorithm (cf. Prop. 2.7):

Input

$F\in \mathbf{F}[X,Y,Z]_d$, $L$ list of closed places of $C=\{F\!\!\:=\!\!\:0\}$,
non-negative integers $d_{[Q]}$, $[Q]\in L$, s.th. $\mathcal{A}=\sum_{[Q]} d_{[Q]}[Q]$,
integers $g_{[Q]}$, $[Q]\in L$, s.th. $G=\sum_{[Q]} g_{[Q]}[Q]$.

Output

Vector space basis of $\mathcal{L}(G)$ (in terms of rational functions on $C$).

1. Choose $m$ sufficiently large (for instance, according to (3), above).
2. Compute $H_0\in \mathbf{F}[X,Y,Z]_m$, $F \not\vert\, H_0$, such that $N^{\ast}H_0\geq {\mathcal{A}}+G_{+}$. To do so, we compute an $\mathbf{F}$-basis for a vector subspace of
   $$V_0:=\left\{ H\in \mathbf{F}[X,Y,Z]_m\;\big\vert\; F \,\vert\, H \text{ or } N^{\ast}H \geq {\mathcal{A}} +G_{+}\right\}\,,$$
   complementary to $W:=F\cdot \mathbf{F}[X,Y,Z]_{m-d}$ (cf. Section 2.4.4) and choose any element $H_0$ of this basis (for instance, with the minimal number of monomials).

3. Compute the effective divisor $R:=N^{\ast}H_0\!-G-\mathcal{A}$ (cf. Section 2.4.3).
4. Compute an $\mathbf{F}$-basis $H_1,\dots, H_s$ of a vector subspace of
   $$V\,:=\, \left\{ H\in \mathbf{F}[X,Y,Z]_m\;\big\vert\; F \,\vert\, H \text{ or } N^{\ast}H \geq {\mathcal{A}} +R\right\} \,,$$
   complementary to $W=F\cdot \mathbf{F}[X,Y,Z]_{m-d}$ (cf. Section 2.4.4).

5. Return the set of rational functions $\mathcal{B}:=\left\{ \dfrac{H_1}{H_0},\dots, \dfrac{H_s}{H_0}\right\}$.