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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 = | 7 5 7 6 5 |
     | 6 6 9 8 5 |
     | 9 1 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          1 2   28   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + -z  + --x -
                                                                  9      9   
     ------------------------------------------------------------------------
     32    71    68        17 2   29     7    323    161   2   1 2        23 
     --y - --z + --, x*z - --z  - --x - --y - ---z + ---, y  + -z  - 5x - --y
      9     9     9        72     18    36     72     18       4           2 
     ------------------------------------------------------------------------
       5               7 2   167    175    11    821   2    1 2   115     7 
     - -z + 59, x*y + --z  - ---x - ---y - --z + ---, x  - --z  - ---x + --y
       4              72      18     36    72     18       36      9     18 
     ------------------------------------------------------------------------
       17    325   3   77 2   70    35    259    46
     + --z + ---, z  - --z  - --x + --y + ---z + --})
       36     9         6      3     3     6      3

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.34295 seconds
i8 : time C = points(M,R);
     -- used 0.308337 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :