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

              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          6 2       
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + -z  - 5y -
                                                                  5         
     ------------------------------------------------------------------------
     89              6 2        94         2   19 2   15          211       
     --z + 59, x*z + -z  - 5x - --z + 64, y  - --z  + --x - 17y + ---z - 17,
      5              5           5             20      2           20       
     ------------------------------------------------------------------------
           3 2             27         2   3 2         27         3      2
     x*y - -z  - 3x - 8y + --z + 12, x  - -z  - 12x + --z + 20, z  - 18z  +
           5                5             5            5
     ------------------------------------------------------------------------
     101z - 180})

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 3.68944 seconds
i8 : time C = points(M,R);
     -- used 0.603908 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :