remainder -- matrix remainder

Synopsis

Usage:

r = remainder(f,g)
Inputs:
- f, a matrix
- g, a Gröbner basis or a matrix, with the same target as f
Outputs:
- r, the remainder of f upon division by g

This operation is the same as Matrix % GroebnerBasis.

The equation g*q+r == f will hold, where q is the map provided by quotient. The source of f should be a free module.

i1 : R = ZZ[x,y]

o1 = R

o1 : PolynomialRing

i2 : f = random(R^2,R^{2:-1})

o2 = | 8x+y  8x+3y |
     | 3x+7y 3x+7y |

             2      2
o2 : Matrix R  <-- R

i3 : g = vars R ++ vars R

o3 = | x y 0 0 |
     | 0 0 x y |

             2      4
o3 : Matrix R  <-- R

i4 : remainder(f,g)

o4 = 0

             2      2
o4 : Matrix R  <-- R

i5 : f = f + map(target f, source f, id_(R^2))

o5 = | 8x+y+1 8x+3y   |
     | 3x+7y  3x+7y+1 |

             2      2
o5 : Matrix R  <-- R

i6 : remainder(f,g)

o6 = | 1 0 |
     | 0 1 |

             2      2
o6 : Matrix R  <-- R