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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 2 3 8 8 |
     | 9 8 5 9 5 |
     | 3 2 2 8 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           13 2   6 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  + -x
                                                                  105     7 
     ------------------------------------------------------------------------
       12    26    484        71 2   23    3    7    107   2   19 2   6   
     - --y - --z + ---, x*z - --z  - --x - -y + -z + ---, y  + --z  + -x -
        7     3     35        70      7    7    2     35       35     7   
     ------------------------------------------------------------------------
     89         1674        4 2                  26   2    1 2   80    15   
     --y - 7z + ----, x*y - -z  - 5x - 2y + 4z + --, x  - --z  - --x - --y +
      7          35         5                     5       14      7     7   
     ------------------------------------------------------------------------
     5    219   3      2
     -z + ---, z  - 13z  + 46z - 48})
     2     7

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 26.8217 seconds
i8 : time C = points(M,R);
     -- used 2.16014 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :