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

## Synopsis

• Usage:
• Inputs:
• Outputs:
• an ideal, the quadratic closure of I

## Description

The quadratic closure of an ideal in a FreeAlgebra is the ideal generated by the generators of I of degree at most 2. Commonly used with homogDual in the case where the ideal generators are homogeneous of degree greater than 1.

If the input is an FreeAlgebraQuotient, the method is applied to the defining ideal of the quotient ring and the corresponding quotient ring is returned. At the moment, quotients of quotients are not implemented, and the ambient ring of the input FreeAlgebraQuotient is assumed to be a FreeAlgebra.

This method is commonly used in conjunction with homogDual.

 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