weightGrevlex -- transform a weight matrix into a monomial ordering matrix

Synopsis

• Usage:

  wtgrev=weightGrevlex(wtR)

• Inputs:
  • wtR, a weight matrix
• Outputs:
  • wtgrev, a matrix, weight-over-grevlex matrix

Description

i1 : wtR = matrix{{5,6,6},{3,6,0}};

              2        3
o1 : Matrix ZZ  <--- ZZ

i2 : weightGrevlex(wtR) 

o2 = | 5 6 6 |
     | 3 6 0 |
     | 1 0 0 |

              3        3
o2 : Matrix ZZ  <--- ZZ

It is standard in other algebra systems to have a weighted monomial ordering based on one row of weights such as matrix{{5,6,6}} being extended to matrix{{5,6,6},{1,1,0},{1,0,0}}, whereas M2 would extend it to matrix{{5,6,6},{1,1,1},{1,1,0}}. The method here allows for more than one independent row of weights matrix{{5,6,6},{3,6,0}} to be extended to matrix{{5,6,6},{3,6,0},{1,0,0}}. Note that the number of rows necessarily matches the number of (free) variables, those of P, since the rightmost square submatrix defines a monomial ordering on P.