Safe Haskell	None
Language	Haskell2010

Algebra.Algorithms.Groebner.Homogeneous

Synopsis

calcGroebnerBasisAfterHomogenising :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly]
calcGroebnerBasisAfterHomogenisingHilb :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly]
calcGroebnerBasisAfterHomogenisingWith :: (Field (Coefficient poly), IsOrderedPolynomial poly) => (forall poly'. (Field (Coefficient poly'), IsOrderedPolynomial poly') => Ideal poly' -> [poly']) -> Ideal poly -> [poly]
calcHomogeneousGroebnerBasis :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> Maybe [poly]
unsafeCalcHomogeneousGroebnerBasis :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly]
hilbertPoincareSeries :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> HPS (Arity poly)
hilbertPoincareSeriesBy :: forall ord poly. (IsMonomialOrder (Arity poly) ord, Field (Coefficient poly), IsOrderedPolynomial poly) => ord -> Ideal poly -> HPS (Arity poly)
hilbertPoincareSeriesForMonomials :: forall t n. (KnownNat n, Foldable t) => t (Monomial n) -> HPS n
data HPS n
taylorHPS :: HPS n -> [Integer]
toRationalFunction :: KnownNat n => HPS n -> RationalFunction Rational
calcHomogeneousGroebnerBasisHilbert :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly]
calcHomogeneousGroebnerBasisHilbertBy :: (Field (Coefficient poly), IsOrderedPolynomial poly, IsMonomialOrder (Arity poly) ord) => ord -> Ideal poly -> [poly]
calcHomogeneousGroebnerBasisHilbertWithSeries :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> HPS (Arity poly) -> [poly]

Documentation

calcGroebnerBasisAfterHomogenising :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly] Source #

calcGroebnerBasisAfterHomogenisingHilb :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly] Source #

calcGroebnerBasisAfterHomogenisingWith :: (Field (Coefficient poly), IsOrderedPolynomial poly) => (forall poly'. (Field (Coefficient poly'), IsOrderedPolynomial poly') => Ideal poly' -> [poly']) -> Ideal poly -> [poly] Source #

Calculates Groebner basis once homogenise, apply unsafeCalcHomogeneousGroebnerBasis, and then dehomogenise.

calcHomogeneousGroebnerBasis :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> Maybe [poly] Source #

Calculates a Groebner basis of the given homogeneous ideal, i.e. an ideal generated by homogeneous polynomials. Returns Nothing if the given ideal is inhomogeneous.

unsafeCalcHomogeneousGroebnerBasis :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly] Source #

Calculates a Groebner basis of the given homogeneous ideal, i.e. an ideal generated by homogeneous polynomials.

N.B. This function DOES NOT check homogeniety of the given ideal. See also calcHomogeneousGroebnerBasis.

hilbertPoincareSeries :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> HPS (Arity poly) Source #

A variant of hilbertPoincareSeriesBy using Grevlex ordering.

hilbertPoincareSeriesBy :: forall ord poly. (IsMonomialOrder (Arity poly) ord, Field (Coefficient poly), IsOrderedPolynomial poly) => ord -> Ideal poly -> HPS (Arity poly) Source #

Calculates the Hilbert-Poincare serires of a given homogeneous ideal, using the specified monomial ordering.

hilbertPoincareSeriesForMonomials :: forall t n. (KnownNat n, Foldable t) => t (Monomial n) -> HPS n Source #

data HPS n Source #

Instances

Instances details

RightModule Integer (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (*.) :: HPS n -> Integer -> HPS n #
RightModule Natural (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (*.) :: HPS n -> Natural -> HPS n #
LeftModule Integer (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (.*) :: Integer -> HPS n -> HPS n #
LeftModule Natural (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (.*) :: Natural -> HPS n -> HPS n #
LeftModule (Unipol Integer) (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (.*) :: Unipol Integer -> HPS n -> HPS n #
Eq (HPS a) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (==) :: HPS a -> HPS a -> Bool # (/=) :: HPS a -> HPS a -> Bool #
KnownNat n => Show (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods showsPrec :: Int -> HPS n -> ShowS # show :: HPS n -> String # showList :: [HPS n] -> ShowS #
Monoidal (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods zero :: HPS n # sinnum :: Natural -> HPS n -> HPS n # sumWith :: Foldable f => (a -> HPS n) -> f a -> HPS n #
Group (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (-) :: HPS n -> HPS n -> HPS n # negate :: HPS n -> HPS n # subtract :: HPS n -> HPS n -> HPS n # times :: Integral n0 => n0 -> HPS n -> HPS n #
Additive (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous Methods (+) :: HPS n -> HPS n -> HPS n # sinnum1p :: Natural -> HPS n -> HPS n # sumWith1 :: Foldable1 f => (a -> HPS n) -> f a -> HPS n #
Abelian (HPS n) Source #
Instance details Defined in Algebra.Algorithms.Groebner.Homogeneous

taylorHPS :: HPS n -> [Integer] Source #

toRationalFunction :: KnownNat n => HPS n -> RationalFunction Rational Source #

calcHomogeneousGroebnerBasisHilbert :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> [poly] Source #

Calculates homogeneous Groebner basis by Hilbert-driven method, computing Hilbert-Poincare series w.r.t. Grevlex.

calcHomogeneousGroebnerBasisHilbertBy :: (Field (Coefficient poly), IsOrderedPolynomial poly, IsMonomialOrder (Arity poly) ord) => ord -> Ideal poly -> [poly] Source #

First compute Hilbert-Poicare series w.r.t. ord by hilbertPoincareSeriesBy, and then apply calcHomogeneousGroebnerBasisHilbertWithSeries.

calcHomogeneousGroebnerBasisHilbertWithSeries :: (Field (Coefficient poly), IsOrderedPolynomial poly) => Ideal poly -> HPS (Arity poly) -> [poly] Source #