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

              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 2   1   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - -z  - -x -
                                                                  4     2   
     ------------------------------------------------------------------------
          5               3 2   1    33    611    113   2   31 2       325   
     6y + -z + 25, x*z - --z  - -x + --y - ---z - ---, y  - --z  - x - ---y +
          4              76     2    19     76     19       38          19   
     ------------------------------------------------------------------------
     349    1034         5 2        110    71    440   2   17 2         12   
     ---z + ----, x*y + --z  - 4x - ---y - --z + ---, x  - --z  - 13x - --y +
      38     19         38           19    38     19       19           19   
     ------------------------------------------------------------------------
     173    466   3   21 2             49
     ---z + ---, z  - --z  - 3x + 6y + --z - 18})
      19     19        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 = | 0 0 4 9 3 4 3 9 3 5 5 4 9 0 7 9 4 3 8 3 2 9 6 7 4 8 0 9 4 2 0 7 7 4 8
     | 4 9 5 7 3 3 9 3 0 2 7 8 4 8 1 2 2 8 4 6 9 6 8 6 1 9 4 6 0 1 5 9 5 7 6
     | 4 1 3 6 2 5 9 9 3 3 8 3 1 0 6 1 7 3 6 6 2 5 5 3 6 2 3 5 5 4 0 6 2 7 6
     | 2 5 3 0 7 2 4 2 2 6 5 3 2 9 6 7 1 2 5 9 9 6 1 0 6 1 2 8 2 8 8 2 9 9 9
     | 3 3 0 4 0 4 1 5 2 1 5 1 8 9 8 8 0 1 0 4 0 0 7 8 5 4 0 4 4 5 0 7 1 0 6
     ------------------------------------------------------------------------
     3 8 8 8 8 3 1 8 8 9 5 8 8 5 5 1 2 1 8 7 0 8 4 3 6 5 9 6 4 4 4 5 7 4 4 6
     0 8 8 4 2 7 6 1 0 1 3 0 9 7 1 4 1 7 9 9 3 9 9 1 6 7 8 4 5 0 8 9 3 4 7 5
     0 5 1 5 4 2 1 8 8 3 1 0 8 1 9 7 9 9 9 4 4 9 9 0 2 9 8 5 7 2 4 9 1 4 1 1
     9 9 0 2 2 1 3 1 0 8 9 1 2 3 8 2 4 5 1 1 1 4 3 5 1 0 1 2 5 8 6 6 9 5 7 9
     9 5 6 5 6 9 6 5 2 5 3 3 9 5 4 1 3 4 9 3 9 3 4 8 1 2 4 3 6 7 5 7 5 2 0 0
     ------------------------------------------------------------------------
     7 8 7 6 6 5 5 5 0 9 6 4 8 5 9 3 4 9 7 7 5 7 1 7 6 5 9 5 5 9 9 6 8 5 7 5
     9 0 8 2 1 0 5 0 2 2 1 8 7 5 0 8 6 8 6 8 1 9 6 2 3 4 2 8 7 3 0 3 7 6 2 9
     9 4 1 0 0 3 2 9 6 6 4 5 6 1 5 2 6 0 6 4 0 1 0 2 8 1 9 0 6 4 0 8 3 0 1 1
     3 4 6 9 1 9 7 9 8 0 2 3 2 1 9 7 0 2 1 9 5 7 7 9 2 4 0 9 4 9 5 5 5 7 4 0
     8 5 0 4 3 1 4 4 7 1 3 9 7 2 3 3 5 2 7 9 9 0 0 5 7 9 8 9 6 3 7 1 4 8 0 3
     ------------------------------------------------------------------------
     4 1 9 6 3 2 9 8 7 0 2 4 6 0 2 4 9 4 8 8 1 9 0 6 5 6 3 5 1 1 6 5 9 1 9 0
     3 2 4 5 4 4 6 2 4 3 9 8 0 4 2 9 9 2 0 4 1 2 8 5 3 9 8 0 8 5 9 0 7 5 7 0
     2 2 7 1 5 4 9 3 5 0 5 8 4 7 7 4 6 8 2 1 2 8 5 7 5 5 7 4 9 0 3 8 5 3 7 9
     3 5 9 9 8 6 4 9 7 8 4 1 0 3 4 2 2 4 1 0 2 8 0 7 5 8 7 1 5 0 6 0 2 2 4 4
     9 7 0 7 6 2 3 3 6 9 7 1 3 8 8 3 3 4 2 3 1 8 8 1 9 5 5 7 0 2 5 0 6 1 9 3
     ------------------------------------------------------------------------
     6 5 3 7 3 5 3 |
     2 9 8 4 7 8 1 |
     3 4 2 7 4 5 1 |
     6 6 0 4 0 2 2 |
     5 0 3 2 1 4 7 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 21.9866 seconds
i8 : time C = points(M,R);
     -- used 1.98997 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :