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Points :: points

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 = | 2 1 3 3 7 |
     | 8 1 2 1 0 |
     | 8 6 9 9 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          47 2   99 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + --x
                                                                   4      2 
     ------------------------------------------------------------------------
            569               19 2   33    213          2   589 2   1263   
     - 9y + ---z - 477, x*z - --z  + --x + ---z - 171, y  + ---z  - ----x -
             4                 4      2     4                8        4    
     ------------------------------------------------------------------------
          7151    6061        25 2              303          2   1 2   11   
     3y - ----z + ----, x*y - --z  + 53x - 3y + ---z - 510, x  - -z  - --x +
            8       2          2                 2               4      2   
     ------------------------------------------------------------------------
     19         3   81 2          733
     --z - 15, z  - --z  + 105x + ---z - 1062})
      4              2             2

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 11.0714 seconds
i8 : time C = points(M,R);
     -- used 0.925526 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :