minimizeOIGB -- minimize an OI-Gröbner basis

Synopsis

Usage:

minimizeOIGB G
Inputs:
- G, a list,
Optional inputs:
- Verbose => ..., default value false
Outputs:
- a list,

Description

This method minimizes G with respect to the Gröbner property, i.e., removes any elements whose leading monomial is OI-divisible by the leading monomial of another element of G. The Verbose option must be either true or false, depending on whether one wants debug information printed.

i1 : P = makePolynomialOIAlgebra(2, x, QQ);

i2 : F = makeFreeOIModule(e, {1,1,2}, P);

i3 : installGeneratorsInWidth(F, 1);

i4 : installGeneratorsInWidth(F, 2);

i5 : installGeneratorsInWidth(F, 3);

i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);

i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);

i10 : time B = oiGB {b1, b2}
 -- used 0.0359832s (cpu); 0.0378164s (thread); 0s (gc)

o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
        1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
      -----------------------------------------------------------------------
      x   x   x   e           - x   x   x   e          }
       2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3

o10 : List

i11 : use F_3; b3 = x_(2,3)*x_(2,2)*x_(1,1)*e_(3,{2,3},3)-x_(2,1)*x_(1,2)^2*e_(3,{1,3},3);

i13 : C = append(B, b3) -- dump in a redundant element

                                                                           
o13 = {x   e        + x   e       , x   x   e        + x   x   e          ,
        1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
      -----------------------------------------------------------------------
                                                                           
      x   x   x   e           - x   x   x   e          , x   x   x   e     
       2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3   2,3 2,2 1,1 3,{2,
      -----------------------------------------------------------------------
                  2
           - x   x   e          }
      3},3    2,1 1,2 3,{1, 3},3

o13 : List

i14 : minimizeOIGB C -- an element gets removed

                                                                        
o14 = {x   x   e        + x   x   e          , x   x   x   e           -
        1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3   2,3 2,2 1,1 3,{2, 3},3  
      -----------------------------------------------------------------------
           2
      x   x   e          , x   e        + x   e       }
       2,1 1,2 3,{1, 3},3   1,1 1,{1},1    2,1 1,{1},2

o14 : List

Ways to use minimizeOIGB :

minimizeOIGB(List)

For the programmer

The object minimizeOIGB is a method function with options.