projectivePoints -- produces the ideal and initial ideal from the coordinates of a finite set of projective points

Synopsis

Usage:

(inG,G) = projectivePoints(M,R)
Inputs:
- M, a matrix, in which each column consists of the projective coordinates of a point
- R, a ring, homogeneous coordinate ring of the projective space containing the points
Optional inputs:
- VerifyPoints => ..., default value true, Option to projectivePoints.
Outputs:
- inG, an ideal, initial ideal of the set of projective points
- G, a list, list of generators for Grobner basis for ideal of projective points

Description

This function uses a modified Buchberger-Moeller algorithm to compute a grobner basis for the ideal of a finite number of points in projective space.

i1 : R = QQ[x_0..x_2]

o1 = R

o1 : PolynomialRing

i2 : M = random(ZZ^3,ZZ^5)

o2 = | 8 7 3 8 8 |
     | 1 8 7 5 5 |
     | 3 3 8 7 2 |

              3        5
o2 : Matrix ZZ  <--- ZZ

i3 : (inG,G) = projectivePoints(M,R)

              2   3     2     2   79285       204632 2   152667      
o3 = (ideal (x , x , x x ), {x  - -----x x  - ------x  + ------x x  +
              0   1   0 1     0    481  0 1     481  1     481  0 2  
     ------------------------------------------------------------------------
     817272       589744 2   3   318467         187373 2     77360   2  
     ------x x  - ------x , x  + ------x x x  + ------x x  - -----x x  -
       481  1 2     481  2   1    18648 0 1 2    4662  1 2    2331 0 2  
     ------------------------------------------------------------------------
     804017   2   294415 3     2   881693         273316 2     410665   2  
     ------x x  + ------x , x x  - ------x x x  - ------x x  + ------x x  +
      4662  1 2    2331  2   0 1    18648 0 1 2    2331  1 2    4662  0 2  
     ------------------------------------------------------------------------
     2169827   2   1563095 3
     -------x x  - -------x })
       4662  1 2     4662  2

o3 : Sequence

i4 : monomialIdeal G == inG

o4 = true

This algorithm may be faster than computing the intersection of the ideals of each projective point.

Caveat

This function removes zero columns of M and duplicate columns giving rise to the same projective point (which prevent the algorithm from terminating). The user can bypass this step with the option VerifyPoints.

Ways to use `projectivePoints` :

"projectivePoints(Matrix,Ring)"

For the programmer

The object projectivePoints is a method function with options.