A module defining the algebra of commutative polynomials over a field k

- data GlexMonomial v = Glex Int [(v, Int)]
- type GlexPoly k v = Vect k (GlexMonomial v)
- glexVar :: Num k => v -> GlexPoly k v
- class Monomial m where
- bind :: (Monomial m, Num k, Ord b, Show b, Algebra k b) => Vect k (m v) -> (v -> Vect k b) -> Vect k b
- class DivisionBasis b where
- (%%) :: (Fractional k, Ord b, Show b, Algebra k b, DivisionBasis b) => Vect k b -> [Vect k b] -> Vect k b

# Documentation

data GlexMonomial v Source

Monomial GlexMonomial | |

Num k => Coalgebra k (GlexMonomial v) | |

(Num k, Ord v) => Algebra k (GlexMonomial v) | |

Eq v => Eq (GlexMonomial v) | |

Ord v => Ord (GlexMonomial v) | |

Show v => Show (GlexMonomial v) | |

Ord v => DivisionBasis (GlexMonomial v) |

type GlexPoly k v = Vect k (GlexMonomial v)Source

glexVar :: Num k => v -> GlexPoly k vSource

glexVar creates a variable in the algebra of commutative polynomials with Glex term ordering. For example, the following code creates variables called x, y and z:

[x,y,z] = map glexVar ["x","y","z"] :: GlexPoly Q String

bind :: (Monomial m, Num k, Ord b, Show b, Algebra k b) => Vect k (m v) -> (v -> Vect k b) -> Vect k bSource

In effect, we have (Num k, Monomial m) => Monad (v -> Vect k (m v)), with return = var, and (>>=) = bind. However, we can't express this directly in Haskell, firstly because of the Ord b constraint, secondly because Haskell doesn't support type functions.

class DivisionBasis b whereSource

Ord v => DivisionBasis (GlexMonomial v) |

(%%) :: (Fractional k, Ord b, Show b, Algebra k b, DivisionBasis b) => Vect k b -> [Vect k b] -> Vect k bSource

(%%) reduces a polynomial with respect to a list of polynomials.