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

              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   79 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - --y
                                                                  23     23 
     ------------------------------------------------------------------------
       156    127        41 2        15    261    373   2   11 2   210   
     + ---z - ---, x*z - --z  - 8x - --y + ---z + ---, y  - --z  - ---y +
        23     23        46          23     46     23       23      23   
     ------------------------------------------------------------------------
     148    137        59 2        122    637    194   2    3 2          75 
     ---z - ---, x*y - --z  - 7x - ---y + ---z + ---, x  + --z  - 10x + ---y
      23     23        46           23     46     23       46           161 
     ------------------------------------------------------------------------
       201    3471   3   312 2   60    1189    900
     - ---z + ----, z  - ---z  - --y + ----z - ---})
       322     161        23     23     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 = | 2 7 9 2 8 2 7 6 8 1 0 0 4 4 7 0 9 9 4 5 4 2 9 8 8 9 2 0 2 9 6 8 9 5 5
     | 5 1 4 0 8 7 3 9 1 0 4 2 0 8 0 2 7 4 6 1 2 1 6 4 7 2 7 2 1 2 1 2 4 1 2
     | 2 4 2 4 9 8 9 9 1 7 1 5 8 3 8 6 6 2 3 6 5 1 2 3 8 0 7 0 8 2 7 3 0 2 7
     | 9 0 6 1 2 8 9 8 6 5 8 7 5 4 5 2 9 2 5 4 5 9 8 9 9 9 5 0 4 8 2 8 6 4 4
     | 8 2 8 0 9 9 6 9 5 5 5 0 0 0 0 0 9 7 2 0 8 6 9 8 2 5 9 1 7 9 6 4 6 2 3
     ------------------------------------------------------------------------
     7 7 8 1 3 0 4 3 3 1 6 3 6 2 9 6 0 0 8 4 8 7 8 2 6 4 9 8 8 7 1 2 7 5 4 4
     1 1 2 8 2 3 4 9 2 9 8 5 6 2 0 9 5 3 4 5 4 5 0 7 7 2 1 1 8 2 8 3 3 5 7 6
     1 7 4 5 9 4 2 6 1 2 9 5 1 7 6 3 3 8 7 8 9 0 7 7 7 9 9 6 9 1 6 4 6 7 6 9
     8 7 8 6 7 1 7 0 9 0 2 0 7 1 6 8 4 1 5 1 8 5 7 3 5 6 5 2 2 1 4 3 6 0 8 9
     0 4 7 3 6 8 9 9 2 5 0 0 9 2 4 1 9 0 1 5 7 2 8 5 3 0 3 7 9 5 3 7 9 6 0 0
     ------------------------------------------------------------------------
     9 6 4 2 5 6 0 1 6 2 8 7 6 7 6 2 7 2 2 3 7 8 1 1 1 4 4 0 5 2 2 2 1 4 7 2
     6 4 2 1 4 3 3 5 9 3 9 7 2 6 1 7 7 7 0 4 1 7 4 6 0 5 5 1 1 7 9 4 7 9 2 7
     6 4 3 9 8 6 4 5 6 2 4 0 0 2 9 0 2 7 6 5 9 4 4 3 2 6 2 9 6 3 1 4 6 5 7 5
     6 7 3 8 9 6 1 0 5 3 6 2 5 1 0 4 9 8 5 2 0 3 1 0 2 7 8 6 4 1 6 0 1 3 1 3
     3 2 7 6 2 4 0 4 6 9 9 4 5 2 0 7 6 3 7 8 6 3 9 3 0 8 6 3 8 8 4 1 9 1 8 7
     ------------------------------------------------------------------------
     0 5 6 4 5 3 3 8 4 7 6 4 8 2 0 4 2 7 3 0 6 0 0 2 0 5 2 3 9 6 3 3 4 1 2 4
     8 7 7 2 0 9 7 8 7 3 3 3 9 3 5 1 9 5 0 7 4 7 6 7 2 6 2 5 1 0 4 5 8 4 8 5
     9 9 5 5 1 0 7 4 0 3 0 7 1 5 9 5 8 9 9 9 0 1 8 8 8 8 4 1 6 1 1 2 0 4 3 3
     8 8 6 6 2 1 6 7 2 9 7 9 6 4 8 2 8 4 0 8 8 2 8 3 0 6 6 0 5 6 3 7 6 8 5 8
     7 0 6 0 7 2 6 8 2 8 4 4 7 9 3 8 8 1 8 5 6 9 2 0 6 9 7 0 4 5 5 5 6 7 9 2
     ------------------------------------------------------------------------
     5 2 2 3 4 3 2 |
     5 2 5 5 2 0 9 |
     7 7 4 1 7 9 9 |
     7 4 0 7 2 2 9 |
     7 0 0 9 7 7 4 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 12.88 seconds
i8 : time C = points(M,R);
     -- used 1.61 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :