quotientRemainder -- matrix quotient and remainder

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 : (q,r) = quotientRemainder(f,g)

o4 = ({1} | 8 8 |, 0)
      {1} | 1 3 |
      {1} | 3 3 |
      {1} | 7 7 |

o4 : Sequence

i5 : g*q+r == f

o5 = true

i6 : f = f + map(target f, source f, id_(R^2))

o6 = | 8x+y+1 8x+3y   |
     | 3x+7y  3x+7y+1 |

             2       2
o6 : Matrix R  <--- R

i7 : (q,r) = quotientRemainder(f,g)

o7 = ({1} | 8 8 |, | 1 0 |)
      {1} | 1 3 |  | 0 1 |
      {1} | 3 3 |
      {1} | 7 7 |

o7 : Sequence

i8 : g*q+r == f

o8 = true