RingElement % Subring -- Remainder modulo a subring

Synopsis

Operator: %
Usage:

r = f % S
Inputs:
- f, a ring element,
- S, an instance of the type Subring,
Outputs:
- r, a ring element, the normal form of f modulo the subring

Description

The result is zero if and only if the input belongs to the subring. If a sagbi basis is known for the subring then subduction is used to compute the normal forms. If no sagbi basis is known then an extrinsic method is used, similar to groebnerMembershipTest.

i1 : R = QQ[x1, x2, x3];

i2 : S = subring {x1+x2+x3, x1*x2+x1*x3+x2*x3, x1*x2*x3, (x1-x2)*(x1-x3)*(x2-x3)} --usual invariants of A_3

o2 = QQ[p_0..p_3], subring of R

o2 : Subring

i3 : f = x1 + x2 + 2*x3

o3 = x1 + x2 + 2x3

o3 : R

i4 : f % S

o4 = x3

o4 : R

i5 : g = x1^2*x2 + x2^2*x3 + x3^2*x1

       2       2          2
o5 = x1 x2 + x2 x3 + x1*x3

o5 : R

i6 : g % S

o6 = 0

o6 : R

Ways to use this method: