HaskellForMaths-0.4.8: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.CommutativeAlgebra.GroebnerBasis

Description

A module providing an efficient implementation of the Buchberger algorithm for calculating the (reduced) Groebner basis for an ideal, together with some straightforward applications.

Synopsis

# Documentation

sPoly :: (Monomial b, Algebra k b, Ord b, Fractional k, Eq k) => Vect k b -> Vect k b -> Vect k b Source

isGB :: (Monomial m, Algebra k m, Ord m, Fractional k, Eq k) => [Vect k m] -> Bool Source

gb1 :: (Monomial m, Algebra k m, Ord m, Fractional k, Eq k) => [Vect k m] -> [Vect k m] Source

pairWith :: (a1 -> a1 -> a) -> [a1] -> [a] Source

reduce :: (Monomial m, Algebra k m, Ord m, Ord k, Fractional k) => [Vect k m] -> [Vect k m] Source

gb2 :: (Monomial m, Algebra k m, Ord m, Ord k, Fractional k) => [Vect k m] -> [Vect k m] Source

(!) :: [a] -> Int -> a Source

gb2a :: (Monomial m, Algebra k m, Ord m, Ord k, Fractional k) => [Vect k m] -> [Vect k m] Source

gb3 :: (Monomial m, Algebra k m, Ord m, Ord k, Fractional k) => [Vect k m] -> [Vect k m] Source

gb4 :: (Monomial m, Algebra k m, Ord m, Ord k, Fractional k) => [Vect k m] -> [Vect k m] Source

mergeBy :: (a -> a -> Ordering) -> [a] -> [a] -> [a] Source

gb :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] Source

Given a list of polynomials over a field, return a Groebner basis for the ideal generated by the polynomials.

sugar :: (Monomial m2, Monomial m1, Monomial m) => Vect b m1 -> Vect b1 m2 -> m -> Int Source

cmpNormal :: (Ord t5, Ord t4) => ((t, t4), (t1, t5)) -> ((t2, t4), (t3, t5)) -> Ordering Source

cmpSug :: (Ord t4, Ord t3, Ord t2, Num t2) => ((t2, t3), (t, t4)) -> ((t2, t3), (t1, t4)) -> Ordering Source

memberGB :: (Monomial m, Algebra k m, Ord m, Fractional k, Eq k) => Vect k m -> [Vect k m] -> Bool Source

memberI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => Vect k m -> [Vect k m] -> Bool Source

`memberI f gs` returns whether f is in the ideal generated by gs

sumI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] Source

Given ideals I and J, their sum is defined as I+J = {f+g | f <- I, g <- J}.

If fs and gs are generators for I and J, then `sumI fs gs` returns generators for I+J.

The geometric interpretation is that the variety of the sum is the intersection of the varieties, ie V(I+J) = V(I) intersect V(J)

productI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] Source

Given ideals I and J, their product I.J is the ideal generated by all products {f.g | f <- I, g <- J}.

If fs and gs are generators for I and J, then `productI fs gs` returns generators for I.J.

The geometric interpretation is that the variety of the product is the union of the varieties, ie V(I.J) = V(I) union V(J)

intersectI :: (Fractional k, Ord k, Monomial m, Ord m) => [Vect k m] -> [Vect k m] -> [Vect k m] Source

The intersection of ideals I and J is the set of all polynomials which belong to both I and J.

If fs and gs are generators for I and J, then `intersectI fs gs` returns generators for the intersection of I and J

The geometric interpretation is that the variety of the intersection is the union of the varieties, ie V(I intersect J) = V(I) union V(J).

The reason for prefering the intersection over the product is that the intersection of radical ideals is radical, whereas the product need not be.

toElimFst :: (Mon b, Functor f) => f a -> f (Elim2 a b) Source

toElimSnd :: (Mon a, Functor f) => f b -> f (Elim2 a b) Source

isElimFst :: (Mon b, Eq b) => Vect b1 (Elim2 b t) -> Bool Source

fromElimSnd :: Functor f => f (Elim2 t b) -> f b Source

eliminateFst :: (Monomial t, Monomial b, Ord b1, Ord t, Ord b, Fractional b1) => [Vect b1 (Elim2 t b)] -> [Vect b1 b] Source

quotientI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] Source

Given ideals I and J, their quotient is defined as I:J = {f | f <- R, f.g is in I for all g in J}.

If fs and gs are generators for I and J, then `quotientI fs gs` returns generators for I:J.

The ideal quotient is the algebraic analogue of the Zariski closure of a difference of varieties. V(I:J) contains the Zariski closure of V(I)-V(J), with equality if k is algebraically closed and I is a radical ideal.

quotientP :: (Monomial b, Algebra k b, Ord b, Ord k, Fractional k) => [Vect k b] -> Vect k b -> [Vect k b] Source

eliminate :: (Eq k, Fractional k, Ord k, MonomialConstructor m, Monomial (m v), Ord (m v)) => [Vect k (m v)] -> [Vect k (m v)] -> [Vect k (m v)] Source

`eliminate vs gs` returns the elimination ideal obtained from the ideal generated by gs by eliminating the variables vs.

mbasis :: (Ord t, Num t) => [t] -> [t] Source

mbasisQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m] Source

Given variables vs, and a Groebner basis gs, `mbasisQA vs gs` returns a monomial basis for the quotient algebra k[vs]/<gs>. For example, `mbasisQA [x,y] [x^2+y^2-1]` returns a monomial basis for k[x,y]/<x^2+y^2-1>. In general, the monomial basis is likely to be infinite.

ltIdeal :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] Source

Given an ideal I, the leading term ideal lt(I) consists of the leading terms of all elements of I. If I is generated by gs, then `ltIdeal gs` returns generators for lt(I).

hilbertFunQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> Int -> Integer Source

Given variables vs, and a homogeneous ideal gs, `hilbertFunQA vs gs` returns the Hilbert function for the quotient algebra k[vs]/<gs>. Given an integer i, the Hilbert function returns the number of degree i monomials in a basis for k[vs]/<gs>. For a homogeneous ideal, this number is independent of the monomial ordering used (even though the elements of the monomial basis themselves are dependent on the ordering).

If the ideal I is not homogeneous, then R/I is not graded, and the Hilbert function is not well-defined. Specifically, the number of degree i monomials in a basis is likely to depend on which monomial ordering you use.

hilbertSeriesQA1 :: (Monomial m, Algebra k m, Ord m, Ord k, Fractional k) => [Vect k m] -> [Vect k m] -> [Int] Source

hilbertSeriesQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Integer] Source

Given variables vs, and a homogeneous ideal gs, `hilbertSeriesQA vs gs` returns the Hilbert series for the quotient algebra k[vs]/<gs>. The Hilbert series should be interpreted as a formal power series where the coefficient of t^i is the Hilbert function evaluated at i. That is, the i'th element in the series is the number of degree i monomials in a basis for k[vs]/<gs>.

hilbertSeriesQA' :: (Fractional k, Ord k, MonomialConstructor m, Ord (m v), Monomial (m v), Algebra k (m v)) => [Vect k (m v)] -> [Integer] Source

In the case where every variable v occurs in some generator g of the homogeneous ideal (the usual case), then the vs can be inferred from the gs. `hilbertSeriesQA' gs` returns the Hilbert series for the quotient algebra k[vs]/<gs>.

hilbertPolyQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> GlexPoly Q String Source

For i >> 0, the Hilbert function becomes a polynomial in i, called the Hilbert polynomial.

hilbertPolyQA' :: (Fractional k, Ord k, MonomialConstructor m, Ord (m v), Monomial (m v), Algebra k (m v)) => [Vect k (m v)] -> GlexPoly Q String Source

dim :: (Monomial m, Algebra k m, Ord m, Ord k, Fractional k) => [Vect k m] -> [Vect k m] -> Int Source

dim' :: (Monomial (m v), MonomialConstructor m, Algebra k (m v), Ord (m v), Ord k, Fractional k) => [Vect k (m v)] -> Int Source