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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 1 2 6 3 |
     | 9 5 1 1 5 |
     | 2 8 2 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          1 2       
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - -z  + 3x -
                                                                  4         
     ------------------------------------------------------------------------
     17    1    7        13 2   23    45    93    395   2   7 2        17   
     --y - -z - -, x*z + --z  - --x + --y - --z + ---, y  - -z  - 2x - --y +
      4    2    4        16      4    16     8     16       6           2   
     ------------------------------------------------------------------------
           59        25 2   9    43    51    215   2   103 2   65    45   
     13z - --, x*y - --z  - -x - --y + --z - ---, x  - ---z  - --x - --y +
            6        24     2     8     4     24        96      8    32   
     ------------------------------------------------------------------------
     173    353   3      2
     ---z - ---, z  - 14z  + 56z - 64})
      16     96

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.44369 seconds
i8 : time C = points(M,R);
     -- used 0.614932 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :