# groebnerMembershipTest -- Extrinsic method for subring membership

## Synopsis

• Usage:
result = groebnerMembershipTest(f, S)
• Inputs:
• f, ,
• S, an instance of the type Subring,
• Outputs:
• result, , f belongs to S.

## Description

Uses the extrinsic method to test membership of a polynomial in a subring. Note that S must be a subring of a polynomial ring.

 i1 : R = QQ[x,y]; i2 : S = subring {x^2 - x*y, x^4}; i3 : f1 = x^2*y^2 - 2*x^3*y; i4 : groebnerMembershipTest(f1, S) o4 = true i5 : f2 = x^10; i6 : groebnerMembershipTest(f2, S) o6 = false

If a sagbi basis is known for S then it is recommended to use RingElement % Subring.