HaskellForMaths-0.4.7: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Algebras.Commutative

Description

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

Most users should probably use Math.CommutativeAlgebra.Polynomial instead, which is basically the same thing but more fully-featured. This module will probably be deprecated at some point, but remains for now because it has a simpler implementation which may be more helpful for people wanting to understand the code.

Synopsis

# Documentation

data GlexMonomial v Source

Constructors

 Glex Int [(v, Int)]

Instances

 Monomial GlexMonomial (Eq k, Num k) => Coalgebra k (GlexMonomial v) (Eq k, 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 v Source

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`

class Monomial m where Source

Methods

var :: v -> Vect Q (m v) Source

powers :: m v -> [(v, Int)] Source

Instances

 Monomial GlexMonomial Monomial SL2

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

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.

lt :: Vect t t1 -> (t1, t) Source

class DivisionBasis b where Source

Methods

dividesB :: b -> b -> Bool Source

divB :: b -> b -> b Source

Instances

 Ord v => DivisionBasis (GlexMonomial v)

dividesT :: DivisionBasis b => (b, t) -> (b, t1) -> Bool Source

divT :: (DivisionBasis t, Fractional t1) => (t, t1) -> (t, t1) -> (t, t1) Source

quotRemMP :: (DivisionBasis b, Algebra k b, Show b, Ord b, Fractional k, Eq k) => Vect k b -> [Vect k b] -> ([Vect k b], Vect k b) Source

(%%) :: (Eq k, Fractional k, Ord b, Show b, Algebra k b, DivisionBasis b) => Vect k b -> [Vect k b] -> Vect k b infixl 7 Source

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