Polynom-0.1: A commutative algebra package Contents Index

Algebra.GroebnerBasis

Synopsis

newtype GroebnerIdeal r m = MakeGroebnerIdeal [MonoidRing r m] syzygy_polynomial :: (Fractional r, GroebnerMonoid m) => MonoidRing r m -> MonoidRing r m -> MonoidRing r m simple_division :: (Fractional r, GroebnerMonoid m) => (MonoidRing r m, m) -> (MonoidRing r m, m) -> MonoidRing r m multiv_division :: (Fractional r, GroebnerMonoid m) => (MonoidRing r m, m) -> [(MonoidRing r m, m)] -> MonoidRing r m normal_form :: (Fractional r, GroebnerMonoid m) => (MonoidRing r m, m) -> [(MonoidRing r m, m)] -> MonoidRing r m reduced_syzygy_pol :: (Fractional r, GroebnerMonoid m) => MonoidRing r m -> MonoidRing r m -> [(MonoidRing r m, m)] -> MonoidRing r m reduce_groebner_basis :: (Fractional r, GroebnerMonoid m) => GroebnerIdeal r m -> GroebnerIdeal r m cleanup_nonminimal :: (Fractional r, GroebnerMonoid m) => [(MonoidRing r m, m)] -> [(MonoidRing r m, MonoidRing r m)] -> ([(MonoidRing r m, m)], [(MonoidRing r m, MonoidRing r m)]) difficulty :: (Num r, GroebnerMonoid m) => MonoidRing r m -> MonoidRing r m -> m process_syzygies :: (Fractional r, GroebnerMonoid m) => [(MonoidRing r m, m)] -> [(MonoidRing r m, MonoidRing r m)] -> [(MonoidRing r m, m)] groebnerize :: (Fractional r, GroebnerMonoid m) => Ideal (MonoidRing r m) -> GroebnerIdeal r m check_groebnerness :: (Fractional r, GroebnerMonoid m) => GroebnerIdeal r m -> Maybe (GroebnerIdeal r m)

Documentation

newtype GroebnerIdeal r m

Constructors MakeGroebnerIdeal [MonoidRing r m] Instances (Num r, GoodMonoid m) => Show (GroebnerIdeal r m)

syzygy_polynomial :: (Fractional r, GroebnerMonoid m) => MonoidRing r m -> MonoidRing r m -> MonoidRing r m

This computes the S-polynomial of two polynomials.

simple_division :: (Fractional r, GroebnerMonoid m) => (MonoidRing r m, m) -> (MonoidRing r m, m) -> MonoidRing r m

simple_division tries to divide the leading term.

multiv_division :: (Fractional r, GroebnerMonoid m) => (MonoidRing r m, m) -> [(MonoidRing r m, m)] -> MonoidRing r m

normal_form :: (Fractional r, GroebnerMonoid m) => (MonoidRing r m, m) -> [(MonoidRing r m, m)] -> MonoidRing r m

reduced_syzygy_pol :: (Fractional r, GroebnerMonoid m) => MonoidRing r m -> MonoidRing r m -> [(MonoidRing r m, m)] -> MonoidRing r m

reduce_groebner_basis :: (Fractional r, GroebnerMonoid m) => GroebnerIdeal r m -> GroebnerIdeal r m

cleanup_nonminimal :: (Fractional r, GroebnerMonoid m) => [(MonoidRing r m, m)] -> [(MonoidRing r m, MonoidRing r m)] -> ([(MonoidRing r m, m)], [(MonoidRing r m, MonoidRing r m)])

difficulty :: (Num r, GroebnerMonoid m) => MonoidRing r m -> MonoidRing r m -> m

process_syzygies :: (Fractional r, GroebnerMonoid m) => [(MonoidRing r m, m)] -> [(MonoidRing r m, MonoidRing r m)] -> [(MonoidRing r m, m)]

The process_syzygies function takes a list of generators and a list of pairs whose syzygies should be added to the Grbner basis. Syzygies are supposed to come ordered by diffficulty.

groebnerize :: (Fractional r, GroebnerMonoid m) => Ideal (MonoidRing r m) -> GroebnerIdeal r m

check_groebnerness :: (Fractional r, GroebnerMonoid m) => GroebnerIdeal r m -> Maybe (GroebnerIdeal r m)

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