Reduction in subrings -- Remainder modulo a subring

Synopsis

Usage:

r = f % S

R = M % S

r = f % SB

R = M % SB
Inputs:
- f, a ring element,
- M, a matrix,
- S, a subring,
- SB, a SAGBIBasis,
Outputs:
- r, a ring element, the normal form of f modulo the Subring or SAGBIBasis
- R, a matrix, the normal form of entries of M modulo the subring or the SAGBIBasis

The result is zero if and only if the input belongs to the subring. If a subalgebra basis is known for the subring, then subduction is used to compute the normal forms. If no subalgebra 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)}

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

Reduction in subrings -- Remainder modulo a subring

Synopsis

See also