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Let Monom denote the monomial -basis for , resp. the corresponding vector of monomials. We represent a homogeneous form by the vector such that Monom. To compute -bases of adjoint forms as needed by the Brill-Noether algorithm, we apply the following algorithm:

Input
, list of closed places of (defined by ), a positive integer,
non-negative integers , , s.th. ,
non-negative integers , , s.th. .

Output
-basis of a subspace of

complementary to (given in form of a matrix of coefficients w.r.t. Monom).

1. The subspace of forms divisible by . Compute the matrix with    Monom   Monom.

Note that , .

2. The space . For each closed place with do the following:
• compute a matrix such that

where is the primitive parametrization as computed from the sHNE of (up to degree ).
• Let be the degree of the place and let denote the image of after applying times the Frobenius map over . Then compute

row-red NF

Finally, concatenate the to obtain with and compute as the kernel of , that is

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

Next: Computing a basis for Up: Computational Aspects Previous: Computing the divisors
Christoph Lossen
2001-03-21