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Points :: points

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 = | 6 8 0 1 5 |
     | 0 5 6 7 0 |
     | 1 8 1 7 5 |

              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          70 2   44 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + --x
                                                                   9      3 
     ------------------------------------------------------------------------
       41    151    1175          2                           2   5 2   46   
     + --y + ---z - ----, x*z + 9z  - 25x - 24y - 65z + 200, y  - -z  + --x +
        3     3       9                                           2      7   
     ------------------------------------------------------------------------
     4    233    375        5 2   74    74    247    550   2   37 2   289   
     -y + ---z - ---, x*y + -z  - --x - --y - ---z + ---, x  + --z  - ---x -
     7     14     7         2      7     7     14     7        18      21   
     ------------------------------------------------------------------------
     163    547    3625   3   32 2                   80
     ---y - ---z + ----, z  - --z  - 8x - 8y + 31z + --})
      21     42     63         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 = | 4 7 9 1 4 8 5 0 4 1 9 9 9 1 8 1 9 8 7 7 1 2 4 2 8 5 0 3 9 9 2 7 9 8 9
     | 0 7 3 8 4 5 1 7 3 4 4 7 4 9 7 6 1 6 7 3 3 0 1 1 6 8 4 9 3 8 8 1 4 1 1
     | 2 1 8 1 9 8 6 1 1 6 9 3 6 6 6 1 5 6 9 3 8 1 8 1 9 9 5 4 2 4 7 7 3 7 4
     | 1 4 1 2 7 4 1 1 4 0 1 4 0 6 7 3 1 8 7 4 7 7 6 7 2 7 9 9 8 7 4 5 9 6 6
     | 1 3 1 2 7 5 0 6 2 2 5 7 6 0 6 1 9 4 5 3 0 5 1 5 1 3 0 6 3 8 4 6 9 9 1
     ------------------------------------------------------------------------
     2 4 6 9 9 9 5 4 7 5 2 4 0 2 2 4 0 3 6 9 9 4 2 0 4 1 1 3 2 0 0 7 9 0 0 2
     6 2 2 9 2 0 3 7 1 8 9 2 1 7 9 8 0 9 5 7 6 9 1 8 0 8 2 2 4 6 9 1 3 9 9 4
     5 2 7 9 5 1 9 4 9 5 3 2 5 1 6 5 6 8 9 2 7 4 8 7 5 0 4 2 2 0 1 1 0 7 9 9
     3 8 8 7 1 1 4 1 2 0 1 1 0 1 0 9 9 6 1 3 3 4 2 5 3 1 1 3 2 5 6 8 5 1 6 2
     8 6 2 5 5 6 4 7 1 9 1 8 4 9 8 6 8 2 7 5 2 0 8 5 9 2 6 7 0 2 1 2 5 9 3 3
     ------------------------------------------------------------------------
     9 5 7 5 6 3 5 7 4 9 2 9 3 1 6 2 1 7 5 0 1 1 1 0 6 0 7 4 9 7 6 0 3 1 7 8
     2 4 7 2 8 8 2 8 7 3 4 4 1 2 9 2 2 7 3 7 4 1 8 5 7 6 6 4 1 2 6 6 5 9 3 2
     0 1 4 5 2 6 5 3 9 4 2 0 2 4 9 7 3 7 7 8 9 2 8 8 8 3 0 2 4 7 9 6 7 8 1 1
     3 0 3 0 4 0 1 9 8 5 5 6 5 3 7 9 4 3 9 1 3 5 0 1 1 2 9 7 5 0 1 5 8 9 6 2
     6 3 4 4 8 3 8 8 6 2 2 6 5 2 4 7 5 3 9 4 8 3 0 7 8 6 6 4 5 8 9 3 1 2 8 3
     ------------------------------------------------------------------------
     8 3 7 2 3 0 5 9 7 1 4 1 1 1 0 8 8 0 8 2 8 0 3 4 3 0 4 7 9 8 1 2 6 0 4 3
     2 7 4 5 8 2 9 3 9 0 2 6 4 6 8 3 8 5 7 6 2 7 8 7 9 3 4 3 8 2 4 2 8 8 7 0
     6 6 7 8 5 6 6 0 3 0 7 0 2 8 6 9 8 8 3 2 7 8 9 4 7 9 5 4 1 6 7 6 1 5 9 5
     5 7 5 0 8 6 3 6 5 9 4 1 4 5 9 8 6 7 6 3 0 0 5 1 7 7 2 9 1 6 4 2 5 8 2 9
     7 8 9 2 6 0 9 9 9 6 0 7 3 2 5 4 5 7 0 0 5 9 2 4 7 3 0 0 6 3 4 4 7 7 6 9
     ------------------------------------------------------------------------
     3 3 7 9 8 8 4 |
     8 4 5 5 8 0 7 |
     3 8 5 1 0 6 9 |
     2 6 9 6 7 3 4 |
     0 9 2 4 7 1 5 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.42835 seconds
i8 : time C = points(M,R);
     -- used 0.389684 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :