A module providing an efficient implementation of the Buchberger algorithm for calculating the (reduced) Groebner basis for an ideal, together with some straightforward applications.
- gb :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m]
- memberI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => Vect k m -> [Vect k m] -> Bool
- sumI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m]
- productI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m]
- intersectI :: (Fractional k, Ord k, Monomial m, Ord m) => [Vect k m] -> [Vect k m] -> [Vect k m]
- quotientI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m]
- eliminate :: (Fractional k, Ord k, MonomialConstructor m, Monomial (m v), Ord (m v)) => [Vect k (m v)] -> [Vect k (m v)] -> [Vect k (m v)]
- mbasisQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Vect k m]
- ltIdeal :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m]
- hilbertFunQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> Int -> Integer
- hilbertSeriesQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> [Integer]
- hilbertPolyQA :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => [Vect k m] -> [Vect k m] -> GlexPoly Q String
Documentation
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.
memberI :: (Fractional k, Ord k, Monomial m, Ord m, Algebra k m) => Vect k m -> [Vect k m] -> BoolSource
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.
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.
eliminate :: (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.
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 -> IntegerSource
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.
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>.