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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 = | 7 8 6 2 4 |
     | 5 3 0 8 5 |
     | 8 5 1 9 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          29 2   332 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  + ---x
                                                                  23      23 
     ------------------------------------------------------------------------
       381    712    1309        183 2   762    940    2879    3086   2  
     + ---y - ---z - ----, x*z - ---z  - ---x - ---y + ----z + ----, y  +
        23     23     23          46      23     23     46      23       
     ------------------------------------------------------------------------
     45 2   420    431    855    1710        157 2   633    908    2661   
     --z  + ---x + ---y - ---z - ----, x*y - ---z  - ---x - ---y + ----z +
     23      23     23     23     23          46      23     23     46    
     ------------------------------------------------------------------------
     2546   2   145 2   209    688    2433    938   3   240 2   612    768   
     ----, x  + ---z  + ---x + ---y - ----z - ---, z  - ---z  + ---x + ---y -
      23         46      23     23     46      23        23      23     23   
     ------------------------------------------------------------------------
     155    3300
     ---z - ----})
      23     23

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.91725 seconds
i8 : time C = points(M,R);
     -- used 0.478927 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :