oiGB -- compute a Gröbner basis for a submodule of a free OI-module

Synopsis

Usage:

oiGB L
Inputs:
- L, a list,
Optional inputs:
- Strategy => ..., default value Minimize
- Verbose => ..., default value false
Outputs:
- a list,

Description

Given a list of elements L belonging to a free OI-module $\mathbf{F}$, this method computes a Gröbner basis for the submodule generated by L with respect to the monomial order of $\mathbf{F}$. The Verbose option must be either true or false, depending on whether one wants debug information printed.

The Strategy option has the following permissible values:

Strategy => FastNonminimal for no post-processing of the basis
Strategy => Minimize to minimize the basis after it is computed; see minimizeOIGB
Strategy => Reduce to reduce the basis after it is computed; see reduceOIGB

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

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

i3 : installGeneratorsInWidth(F, 1);

i4 : installGeneratorsInWidth(F, 2);

i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);

i7 : 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);

i9 : time oiGB {b1, b2}
 -- used 0.021s (cpu); 0.020874s (thread); 0s (gc)

o9 = {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

o9 : List

Ways to use oiGB:

oiGB(List)

For the programmer

The object oiGB is a method function with options.