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

              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          3    15   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + -x - --y -
                                                                  4     4   
     ------------------------------------------------------------------------
     17               2   5    7    41         2   15    63    15           
     --z + 12, x*z + z  - -x - -y - --z + 20, y  + --x - --y + --z + 20, x*y
      4                   4    4     4              4     4     4           
     ------------------------------------------------------------------------
                           2    2   7    49    65        3      2   15    35 
     + x - 14y + 6z + 16, x  - z  - -x - --y + --z - 4, z  - 14z  + --x - --y
                                    4     4     4                    8     8 
     ------------------------------------------------------------------------
       463
     + ---z - 59})
        8

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.73661 seconds
i8 : time C = points(M,R);
     -- used 0.722858 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :