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

              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          41 2   1   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + -x -
                                                                  24     4   
     ------------------------------------------------------------------------
     33    115    187        65 2   39     7    123    67   2   17 2       
     --y + ---z - ---, x*z - --z  - --x + --y + ---z - --, y  + --z  - 2x -
      8     8      6         48      8    16     16    12        3         
     ------------------------------------------------------------------------
                628        2 2                   148   2   17 2   19    7   
     9y - 67z + ---, x*y + -z  - 2x - 8y - 10z + ---, x  + --z  - --x - -y -
                 3         3                      3         4      2    4   
     ------------------------------------------------------------------------
     217          3      2
     ---z + 177, z  - 17z  + 92z - 160})
      4

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.00023 seconds
i8 : time C = points(M,R);
     -- used 0.405545 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :