forceGB -- declare that the columns of a matrix are a Gröbner basis

Synopsis

Usage:

forceGB f
Inputs:
- f, a matrix
Optional inputs:
- ChangeMatrix => ..., default value null, inform Macaulay2 about the change of basis matrix from GB to generators
- MinimalMatrix => ..., default value null, specify the minimal generator matrix
- SyzygyMatrix => ..., default value null, specify the syzygy matrix
Outputs:
- a Gröbner basis

Description

Declares that the columns of the matrix f constitute a Gröbner basis, autoreduces it, minimizes it, sorts it, and returns a Gröbner basis object declaring itself complete, without computing any S-pairs.

Sometimes one knows that a set of polynomials (or columns of such) form a Gröbner basis, but Macaulay2 doesn't. This is the way to inform the system of this fact.

i1 : gbTrace = 3;

i2 : R = ZZ[x,y,z];

   -- registering polynomial ring 3 at 0x7f7b18d5c400

i3 : f = matrix{{x^2-3, y^3-1, z^4-2}};

             1       3
o3 : Matrix R  <--- R

i4 : g = forceGB f

o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 0]

o4 : GroebnerBasis

This Gröbner basis object is stored with the matrix and can be obtained as usual:

i5 : g === gb(f, StopBeforeComputation=>true)

o5 = true

Requesting a Gröbner basis for f requires no computation.

i6 : gens gb f

o6 = | x2-3 y3-1 z4-2 |

             1       3
o6 : Matrix R  <--- R

If an autoreduced Gröbner basis is desired, replace f by gens forceGB f first.

Caveat

If the columns do not form a Gröbner basis, nonsensical answers may result

Ways to use `forceGB` :

"forceGB(Matrix)"

For the programmer

The object forceGB is a method function with options.