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homogDual -- Computes the dual of a pure homogeneous ideal

Synopsis

Description

The homogeneous dual of a pure ideal I in an NCPolynomialRing A is generated by the orthogonal complement to the generators of I under the natural pairing on the generating subspace of A and its linear dual. Though technically the dual ideal belongs to the tensor algebra on the dual space of generators, this method returns the dual ideal in the same NCPolynomialRing.

If the input is an NCQuotient ring, the method is applied to the defining ideal of the quotient and the corresponding quotient ring is returned.

Commonly used in conjunction with quadraticClosure.

i1 : A = QQ{x,y,z}

o1 = A

o1 : NCPolynomialRing
 
i2 : I = ncIdeal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}

                                    2    2     3  3
o2 = Two-sided ideal {-zx+xz, yz, -y x+xy , -yx +x y}

o2 : NCIdeal
 
i3 : J = quadraticClosure I

o3 = Two-sided ideal {-zx+xz, yz}

o3 : NCIdeal
 
i4 : J' = homogDual J

                       2           2              2
o4 = Two-sided ideal {x , yx, xy, y , zy, zx+xz, z }

o4 : NCIdeal

See also

Ways to use homogDual :

For the programmer

The object homogDual is a method function.