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

Synopsis

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 = | 2 4 0 6 1 |
     | 2 6 8 2 3 |
     | 1 0 0 3 2 |

              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          8 2   2   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + -z  - -x -
                                                                  3     3   
     ------------------------------------------------------------------------
     4    34    32           2                        2   44 2   17    8   
     -y - --z + --, x*z - 10z  + 2x + 4y + 28z - 32, y  - --z  + --x - -y +
     3     3     3                                         3      3    3   
     ------------------------------------------------------------------------
     142    128        52 2   31    26    158    208   2     2            
     ---z - ---, x*y + --z  - --x - --y - ---z + ---, x  - 8z  - 3x + 2y +
      3      3          3      3     3     3      3                       
     ------------------------------------------------------------------------
                3   20 2   2    4    37    32
     22z - 16, z  - --z  + -x + -y + --z - --})
                     3     3    3     3     3

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.181 seconds
i8 : time C = points(M,R);
     -- used 0.882 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :