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

              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          43 2   275 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + ---x
                                                                  18     108 
     ------------------------------------------------------------------------
       107    1481    1025        5 2   187    5    250    265   2   13 2  
     - ---y + ----z - ----, x*z + -z  - ---x - -y - ---z + ---, y  - --z  +
        18     54      12         3      18    3     9      2         6    
     ------------------------------------------------------------------------
     20    25    503              13 2   121    43    503    235   2   2 2  
     --x - --y + ---z - 80, x*y + --z  - ---x - --y - ---z + ---, x  - -z  -
      9     3     18               9      27     9     27     3        3    
     ------------------------------------------------------------------------
     118    4    70         3   56 2   25    10    1003
     ---x - -y + --z + 25, z  - --z  - --x - --y + ----z - 185})
      9     3     9              3      9     3      9

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.71038 seconds
i8 : time C = points(M,R);
     -- used 0.451072 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :