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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 = | 4 7 0 8 8 |
     | 2 4 2 6 7 |
     | 4 5 9 8 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          83 2   20 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - --x
                                                                  27     27 
     ------------------------------------------------------------------------
       263    1009    2318        133 2   223    142    1637    3676   2  
     - ---y + ----z - ----, x*z - ---z  - ---x - ---y + ----z - ----, y  -
        27     27      27          27      27     27     27      27       
     ------------------------------------------------------------------------
     47 2   70    389    667    1526        26 2   44    260    346    488 
     --z  + --x - ---y + ---z - ----, x*y - --z  - --x - ---y + ---z - ---,
     27     27     27     27     27         27     27     27     27     27 
     ------------------------------------------------------------------------
      2   19 2   71    10    275    916   3      2
     x  + --z  - --x + --y - ---z + ---, z  - 15z  + 6y + 62z - 84})
           9      9     9     9      9

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.2885 seconds
i8 : time C = points(M,R);
     -- used 0.377983 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :