Math.Algebras.Commutative

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.

Documentation

data GlexMonomial v

Constructors

Glex Int [(v, Int)]

Instances

Monomial GlexMonomial

Defined in Math.Algebras.Commutative

Methods

var :: v -> Vect Q (GlexMonomial v)

powers :: GlexMonomial v -> [(v, Int)]

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

Defined in Math.Algebras.Commutative

Methods

counit :: Vect k (GlexMonomial v) -> k

comult :: Vect k (GlexMonomial v) -> Vect k (Tensor (GlexMonomial v) (GlexMonomial v))

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

Defined in Math.Algebras.Commutative

Methods

unit :: k -> Vect k (GlexMonomial v)

mult :: Vect k (Tensor (GlexMonomial v) (GlexMonomial v)) -> Vect k (GlexMonomial v)

Eq v => Eq (GlexMonomial v)

Defined in Math.Algebras.Commutative

Methods

(==) :: GlexMonomial v -> GlexMonomial v -> Bool

(/=) :: GlexMonomial v -> GlexMonomial v -> Bool

Ord v => Ord (GlexMonomial v)

Defined in Math.Algebras.Commutative

Methods

compare :: GlexMonomial v -> GlexMonomial v -> Ordering

(<) :: GlexMonomial v -> GlexMonomial v -> Bool

(<=) :: GlexMonomial v -> GlexMonomial v -> Bool

(>) :: GlexMonomial v -> GlexMonomial v -> Bool

(>=) :: GlexMonomial v -> GlexMonomial v -> Bool

max :: GlexMonomial v -> GlexMonomial v -> GlexMonomial v

min :: GlexMonomial v -> GlexMonomial v -> GlexMonomial v

Show v => Show (GlexMonomial v)

Defined in Math.Algebras.Commutative

Methods

showsPrec :: Int -> GlexMonomial v -> ShowS

show :: GlexMonomial v -> String

showList :: [GlexMonomial v] -> ShowS

Ord v => DivisionBasis (GlexMonomial v)

Defined in Math.Algebras.Commutative

Methods

dividesB :: GlexMonomial v -> GlexMonomial v -> Bool

divB :: GlexMonomial v -> GlexMonomial v -> GlexMonomial v

type GlexPoly k v = Vect k (GlexMonomial v)

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

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

Methods

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

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

Instances

Monomial GlexMonomial

Defined in Math.Algebras.Commutative

Methods

var :: v -> Vect Q (GlexMonomial v)

powers :: GlexMonomial v -> [(v, Int)]

Monomial SL2

Defined in Math.Algebras.AffinePlane

Methods

var :: v -> Vect Q (SL2 v)

powers :: SL2 v -> [(v, Int)]

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

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 k b -> (b, k)

class DivisionBasis b where

Methods

dividesB :: b -> b -> Bool

divB :: b -> b -> b

Instances

Ord v => DivisionBasis (GlexMonomial v)

Defined in Math.Algebras.Commutative

Methods

dividesB :: GlexMonomial v -> GlexMonomial v -> Bool

divB :: GlexMonomial v -> GlexMonomial v -> GlexMonomial v

dividesT :: DivisionBasis b1 => (b1, b2) -> (b1, b3) -> Bool

divT :: (DivisionBasis a, Fractional b) => (a, b) -> (a, b) -> (a, b)

quotRemMP :: (DivisionBasis b2, Fractional b1, Eq b1, Ord b2, Show b2, Algebra b1 b2) => Vect b1 b2 -> [Vect b1 b2] -> ([Vect b1 b2], Vect b1 b2)

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

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