Computing $\mathbf{F}$-bases of adjoint forms

Let Monom$(m)=\{G_1,\dots,G_M\}$ denote the monomial $\mathbf{F}$-basis for $\mathbf{F}[X,Y,Z]_m$, resp. the corresponding vector of monomials. We represent a homogeneous form $H\in \mathbf{F}[X,Y,Z]_m$ by the vector $v\in \mathbf{F}^n$ such that $H=$Monom$(m)^T\!\cdot v$. To compute $\mathbf{F}$-bases of adjoint forms as needed by the Brill-Noether algorithm, we apply the following algorithm:

Input

$F\in \mathbf{F}[X,Y,Z]_d$, $L$ list of closed places of $C$ (defined by $F=0$), $m>d$ a positive integer,
non-negative integers $d_{[Q]}$, $[Q]\in L$, s.th. $\mathcal{A}=\sum_{[Q]} d_{[Q]}[Q]$,
non-negative integers $n_{[Q]}$, $[Q]\in L$, s.th. $E=\sum_{[Q]} n_{[Q]}[Q]$.

Output

$\mathbf{F}$-basis of a subspace of

$$V:=\left\{ H\in \mathbf{F}[X,Y,Z]_m\;\big\vert\; F \,\vert\, H \text{ or } N^{\ast}H \geq {\mathcal{A}}+E\right\}$$

complementary to $W:=F\cdot \mathbf{F}[X,Y,Z]_{m-d}$ (given in form of a matrix of coefficients w.r.t. Monom$(m)$).

1. The subspace $W\subset V$ of forms divisible by $F$. Compute the matrix $W\in Mat(\!\:\ell\times\!\!\: M,\mathbf{F}\,)$ with $W \cdot$   Monom$(m)=F\cdot$   Monom$(m\!\!\:-\!\!\:d)$.

   Note that $\ell=\tbinom{m-d+2}{2}$, $M=\tbinom{m+2}{2}$.

2. The space $V$. For each closed place $[Q]$ with $m_{[Q]}:=d_{[Q]}+n_{[Q]}>0$ do the following:
   • compute a matrix $A_Q=(a_{ij})_{i,j}\in Mat(\!\:m_{[Q]}\!\times\!\!\: M,\overline{\mathbf{F}}\,)$ such that
     $$G_j\bigl(\varphi_Q(x),\varphi_Q(y)\bigr) = \sum_{i=0}^{m_{[Q]}-1} a_{ij}t^i\,, \quad j=1,\dots, M\,,$$
     where $\varphi_Q$ is the primitive parametrization as computed from the sHNE of $Q$ (up to degree $m_{[Q]}$).
   • Let $k_Q$ be the degree of the place $Q$ and let $A_Q^{(i)}$ denote the image of $A_Q$ after applying $i$ times the Frobenius map over $\mathbf{F}$. Then compute
     $$A_{[Q]} \,:=\,$$   row-red NF$$\left( \begin{array}{c} A_Q\\ A_Q^{(1)}\\ \vdots\\ A_Q^{(k_Q-1)} \end{array}\right) \in Mat(\!\:k_Qm_{[Q]}\!\times\!\!\: M,\mathbf{F}\,)\,.$$

   Finally, concatenate the $A_{[Q]}$ to obtain $A\in Mat(\!\:K\!\times\!\!\: M,\mathbf{F}\,)$ with $K=\sum_{[Q]} k_Qm_{[Q]}$ and compute $V$ as the kernel of $A$, that is
   $$V:=Ker(A)\in Mat(\!\:k\times\!\!\: M,\mathbf{F}\,)\,, \qquad A\cdot V^T\!=0\,.$$

3. Compute a complement of $W$ in $V$. This can be done, for instance, by using the lift command in SINGULAR.