invReduce -- compute normal form modulo involutive basis by involutive reduction

• Usage:

  (r,c) = invReduce(p,J)

• Inputs:
  • p, a matrix, the columns are to be reduced modulo J
  • J, an instance of the type InvolutiveBasis
• Outputs:
  • r, a matrix, the normal form of (the columns of) p modulo J
  • c, a matrix, the reduction coefficients
• Consequences:

  • the columns of r are in normal form modulo J, and p = r + J#0 * c, where * is matrix multiplication

i1 : R = QQ[x,y,z];

i2 : M = matrix {{x+y+z, x*y+y*z+z*x, x*y*z-1}};

             1      3
o2 : Matrix R  <-- R

i3 : J = janetBasis M;

i4 : p = matrix {{y,y^2,y^3}}

o4 = | y y2 y3 |

             1      3
o4 : Matrix R  <-- R

i5 : invReduce(p,J)

o5 = (| y -yz-z2 1 |, | 0 0 0   |)
                      | 0 1 y-z |
                      | 0 0 1   |
                      | 0 0 0   |

o5 : Sequence