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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 = | 5 1 1 8 0 |
     | 3 3 4 3 7 |
     | 8 9 5 3 7 |

              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           34 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  -
                                                                  217    
     ------------------------------------------------------------------------
      60    1637    313    669        477 2   1341    596    6746    2811   2
     ---x - ----y - ---z + ---, x*z + ---z  - ----x + ---y - ----z + ----, y 
     217     217    217     31        217      217    217     217     31     
     ------------------------------------------------------------------------
        51 2    90    2347    507    678         17 2   621     59    169   
     - ---z  - ---x - ----y + ---z + ---, x*y + ---z  - ---x + ---y - ---z -
       217     217     217    217     31        217     217    217    217   
     ------------------------------------------------------------------------
      9   2   10 2   51    24    134         3   3966 2   660    864   
     --, x  - --z  - --x - --y + ---z - 40, z  - ----z  + ---x + ---y +
     31        7      7     7     7               217     217    217   
     ------------------------------------------------------------------------
     22973    6708
     -----z - ----})
      217      31

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.7603 seconds
i8 : time C = points(M,R);
     -- used 0.852053 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :