# homogDual -- Computes the dual of a pure homogeneous ideal

## Description

The homogeneous dual of a pure (i.e. generators all in the same degree) ideal I in a FreeAlgebra 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 FreeAlgebra

If the input is a FreeAlgebraQuotient 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 : FreeAlgebra i2 : I = ideal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3} 2 2 3 3 o2 = ideal (x*z - z*x, y*z, x*y - y x, x y - y*x ) o2 : Ideal of A i3 : J = quadraticClosure I o3 = ideal (x*z - z*x, y*z) o3 : Ideal of A i4 : J' = homogDual J Using numthreads = 0 2 2 2 o4 = ideal (x , x*y, y*x, y , x*z + z*x, z*y, z ) o4 : Ideal of A

• quadraticClosure -- Creates the subideal generated by quadratic elements of a given ideal

## Ways to use homogDual :

• "homogDual(FreeAlgebra)"
• "homogDual(FreeAlgebraQuotient)"
• "homogDual(Ideal)"

## For the programmer

The object homogDual is .