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

              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          3 2       
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - -z  - 5y +
                                                                  4         
     ------------------------------------------------------------------------
     3    45         2             2    69 2   24    146     93    4635     
     -z + --, x*z - z  - 5x + 5z, y  - ---z  + --x - ---y + ---z + ----, x*y
     2     4                           208     13     13    104     208     
     ------------------------------------------------------------------------
       31 2   62    28    135    81   2    63 2   115    42    129    387   3
     - --z  - --x - --y + ---z + --, x  - ---z  - ---x + --y + ---z - ---, z 
       52     13    13     52    13       104      13    13     52    104
     ------------------------------------------------------------------------
          2
     - 15z  + 59z - 45})

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 3.02954 seconds
i8 : time C = points(M,R);
     -- used 0.611907 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :