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

              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          131 2   27 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  + --x
                                                                  340     34 
     ------------------------------------------------------------------------
       71    175    1399        267 2   14    12    250    728   2   633 2  
     - --y - ---z + ----, x*z - ---z  - --x - --y + ---z - ---, y  + ---z  -
       17     68     85          85     17    17     17     85       170    
     ------------------------------------------------------------------------
      9    151    871    3646        173 2   225     9    1035    343   2  
     --x - ---y - ---z + ----, x*y + ---z  - ---x - --y - ----z + ---, x  +
     17     17     34     85          68      34    17     68      17      
     ------------------------------------------------------------------------
     1297 2   178    32    1739    2338   3      2
     ----z  - ---x + --y - ----z + ----, z  - 11z  + 34z - 24})
      510      17    17     102     85

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 2.67107 seconds
i8 : time C = points(M,R);
     -- used 0.305134 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :