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

Synopsis

Usage:
(Q,inG,G) = points(M,R)
Inputs:
- M, a matrix, in which each column consists of the coordinates of a point
- R, a polynomial ring, coordinate ring of the affine space containing the points
Outputs:
- Q, a list, list of standard monomials
- inG, an ideal, initial ideal of the set of points
- G, a list, list of generators for Grobner basis for ideal of points

Description

This function uses the Buchberger-Moeller algorithm to compute a grobner basis for the ideal of a finite number of points in affine space. Here is a simple example.

i1 : M = random(ZZ^3, ZZ^5)

o1 = | 0 3 6 0 0 |
     | 0 7 7 7 8 |
     | 6 4 7 1 3 |

              3        5
o1 : Matrix ZZ  <--- ZZ

i2 : R = QQ[x,y,z]

o2 = R

o2 : PolynomialRing

i3 : (Q,inG,G) = points(M,R)

                    2                     2        2   3          21    99   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --x - --y -
                                                                  19    19   
     ------------------------------------------------------------------------
     112    672         2   72    30    91    138   2   56    173    56   
     ---z + ---, x*z - z  - --x - --y + --z + ---, y  + --x - ---y - --z +
      19     19             19    19    19     19       19     19    19   
     ------------------------------------------------------------------------
     336             2    2   53    30    91    138   3      2   46    60   
     ---, x*y - 7x, x  - z  - --x - --y + --z + ---, z  - 12z  + --x - --y +
      19                      19    19    19     19              19    19   
     ------------------------------------------------------------------------
     695    66
     ---z - --})
      19    19

o3 : Sequence

i4 : monomialIdeal G == inG

o4 = true

Next a larger example that shows that the Buchberger-Moeller algorithm in points may be faster than the alternative method using the intersection of the ideals for each point.

i5 : R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]

o5 = R

o5 : PolynomialRing

i6 : M = random(ZZ^5, ZZ^150)

o6 = | 6 0 0 5 6 8 1 1 3 9 1 0 0 2 7 9 8 0 5 9 8 4 0 3 9 6 0 8 6 9 9 3 6 1 6
     | 2 1 4 6 2 2 7 8 8 5 5 7 3 4 6 2 2 4 6 3 4 4 7 0 8 6 9 6 2 8 7 8 5 3 8
     | 6 3 5 5 8 2 4 7 7 7 3 2 2 8 0 3 4 2 1 6 7 5 1 6 9 2 0 3 0 8 8 1 1 0 7
     | 1 0 2 7 7 6 7 0 7 7 4 2 2 7 0 5 8 9 3 3 9 3 7 4 3 2 7 5 8 1 7 8 1 6 4
     | 0 6 9 9 3 8 6 0 4 2 3 5 1 5 9 4 0 4 6 1 7 3 8 2 5 1 8 0 7 8 1 5 2 7 7
     ------------------------------------------------------------------------
     5 7 6 8 1 0 6 9 5 3 1 8 7 9 4 2 5 9 3 2 2 2 2 5 8 9 5 4 6 4 0 8 8 0 5 1
     1 3 1 4 4 9 1 9 4 3 8 5 6 7 0 6 4 5 6 7 5 1 3 3 1 6 5 2 7 6 4 2 2 1 5 2
     6 6 3 1 6 2 0 5 1 9 5 4 7 2 1 8 0 5 3 4 2 7 9 9 6 0 4 7 1 5 4 0 7 0 6 9
     1 8 1 4 6 1 4 1 3 2 9 2 5 9 8 0 3 8 5 5 9 3 0 1 1 2 7 1 5 3 3 5 0 2 7 9
     8 6 1 7 5 2 4 3 8 1 5 0 8 4 0 5 4 3 3 3 4 8 6 9 6 8 0 7 4 6 4 8 9 1 4 4
     ------------------------------------------------------------------------
     5 4 7 6 1 5 8 5 6 0 1 0 7 7 2 2 3 1 3 3 5 6 8 5 8 4 3 8 1 4 6 0 9 5 0 4
     7 9 1 1 6 2 9 3 1 1 3 7 4 7 7 2 9 1 5 7 0 1 4 7 1 3 0 1 7 3 5 0 9 3 5 8
     9 2 3 1 8 4 6 6 6 3 0 1 9 8 2 5 7 0 2 0 3 1 2 9 5 5 1 9 8 5 1 4 2 2 7 5
     3 9 9 0 9 6 2 3 2 5 2 5 1 8 5 1 2 5 9 4 3 8 1 9 5 2 2 6 8 4 9 9 1 7 2 1
     0 6 0 6 7 6 1 5 0 9 5 4 2 2 6 0 4 9 5 9 0 5 7 6 1 9 7 9 3 9 1 1 5 2 4 8
     ------------------------------------------------------------------------
     1 9 5 3 7 5 9 2 4 9 9 2 8 3 2 6 8 3 7 4 6 0 9 1 3 8 6 4 1 9 8 4 1 9 2 4
     6 3 5 2 5 6 5 7 5 2 4 3 6 8 5 1 2 7 0 3 1 4 7 9 4 6 6 0 4 8 7 0 4 2 1 2
     2 2 9 1 3 4 2 6 1 1 0 3 4 3 4 7 5 5 5 0 1 3 4 2 2 4 7 9 3 6 8 8 5 4 3 4
     3 3 7 8 5 1 3 3 7 8 7 4 8 4 7 9 9 7 8 5 0 3 3 9 1 7 7 0 6 2 0 5 0 8 4 1
     2 9 0 0 4 3 1 9 4 8 7 3 6 2 6 4 9 7 0 4 2 2 9 7 7 5 7 0 6 2 7 9 6 7 7 7
     ------------------------------------------------------------------------
     1 9 4 5 9 8 0 |
     6 0 3 7 8 6 3 |
     4 6 2 1 2 8 9 |
     4 6 9 8 0 7 3 |
     6 5 5 3 4 4 0 |

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 8.10874 seconds

i8 : time C = points(M,R);
     -- used 1.1245 seconds

i9 : J == C_2  

o9 = true

Ways to use `points` :

points(Matrix,Ring)

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

Synopsis

Description

See also

Ways to use points :

Ways to use `points` :