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

              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           7 2   47 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + --x
                                                                  76     38 
     ------------------------------------------------------------------------
       44    197    223        111 2   622    291    204    303   2   1061 2
     - --y - ---z + ---, x*z + ---z  - ---x + ---y - ---z + ---, y  - ----z 
       19     76     38        133     133    133     19     19       1064  
     ------------------------------------------------------------------------
       943    1949    1819    1035        1569 2   2103    783    2463   
     - ---x - ----y + ----z - ----, x*y - ----z  - ----x - ---y + ----z -
       532     266     152     76         1064      532    266     152   
     ------------------------------------------------------------------------
     1791   2    549 2   3925     39    459    291   3   1863 2   540    150 
     ----, x  + ----z  - ----x + ---y - ---z + ---, z  - ----z  + ---x - ---y
      76        1064      532    266    152     76        133     133    133 
     ------------------------------------------------------------------------
       1034    1134
     + ----z - ----})
        19      19

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 5.59115 seconds
i8 : time C = points(M,R);
     -- used 0.584911 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :