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

              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           8 2   15 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  - --x
                                                                  51     34 
     ------------------------------------------------------------------------
       57    211    729        31 2   96    36    191    912   2   1 2       
     - --y - ---z + ---, x*z + --z  - --x - --y - ---z + ---, y  - -z  - 9y +
       17     34     34        51     17    17     17     17       3         
     ------------------------------------------------------------------------
                    52 2   143    98    171    1401   2    7 2   144    48   
     5z + 2, x*y + ---z  - ---x - --y - ---z + ----, x  + --z  - ---x + --y -
                   153      34    17     34     34        17      17    17   
     ------------------------------------------------------------------------
     74    127   3      2
     --z + ---, z  - 18z  + 99z - 162})
     17     17

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 3.60445 seconds
i8 : time C = points(M,R);
     -- used 0.641902 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :