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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 = | 2 9 8 1 5 |
     | 5 2 5 8 5 |
     | 0 7 1 6 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          13 2    7 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - --x
                                                                  24     12 
     ------------------------------------------------------------------------
       70    23    721        49 2    7    259    709    2611   2   13 2   7 
     - --y - --z + ---, x*z + --z  + --x + ---y - ---z - ----, y  + --z  + -x
        9    24     18        72     36     27     72     54         4     2 
     ------------------------------------------------------------------------
       7    97    61        13 2   29    137    97    769   2   199 2   77   
     - -y - --z - --, x*y - --z  - --x - ---y + --z + ---, x  + ---z  - --x +
       3     4     3         3      3     9      3     9         36     18   
     ------------------------------------------------------------------------
     350    1435    1627   3   161 2   35    70    569    455
     ---y - ----z - ----, z  - ---z  - --x - --y + ---z + ---})
      27     36      27         12      6     9     12     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 = | 5 2 7 9 2 5 0 2 5 7 8 1 3 1 5 5 1 8 2 6 6 6 9 8 1 7 0 2 0 2 4 9 2 5 6
     | 1 5 8 1 8 6 6 9 4 2 2 8 6 8 2 8 0 9 3 3 6 6 9 8 3 4 5 9 8 4 7 3 8 6 0
     | 9 6 1 1 8 8 6 1 2 0 0 6 6 6 0 2 6 8 8 1 3 6 3 9 7 6 6 7 1 0 5 7 8 9 5
     | 1 1 6 6 6 8 8 0 3 4 5 3 7 0 9 5 5 7 3 1 9 6 9 9 4 3 4 2 4 9 4 4 2 3 8
     | 9 0 2 6 3 0 7 6 2 3 2 7 6 6 7 9 6 1 9 1 7 3 8 8 0 3 3 0 8 8 6 1 7 8 6
     ------------------------------------------------------------------------
     3 4 4 7 0 4 4 3 3 1 5 7 7 4 0 4 9 7 7 2 3 1 8 6 7 3 2 4 7 4 3 8 6 6 9 9
     5 0 8 7 4 2 4 1 2 1 0 1 1 7 3 3 7 4 8 9 3 3 7 5 8 0 5 6 7 4 7 2 6 7 9 5
     5 7 7 6 9 0 8 2 8 3 9 3 5 4 7 2 5 2 8 3 5 2 2 7 2 7 7 4 0 1 2 8 6 1 8 0
     0 9 7 6 3 5 6 3 4 3 8 8 2 1 0 2 3 1 2 3 9 9 8 2 2 7 3 7 3 0 2 7 1 4 9 2
     5 2 7 0 3 9 6 1 5 0 1 6 5 9 9 0 1 9 2 1 0 1 9 8 8 1 1 9 3 9 0 1 6 8 5 0
     ------------------------------------------------------------------------
     8 6 0 5 4 4 6 9 4 5 7 8 0 7 1 0 5 8 8 6 5 2 3 4 0 7 2 9 5 4 0 7 3 1 6 9
     2 4 3 1 2 1 2 1 6 0 1 9 1 1 0 5 1 0 7 8 4 8 9 4 2 5 6 6 1 6 1 2 4 0 4 1
     9 9 6 5 7 1 5 2 2 4 4 9 6 5 5 6 3 6 7 7 3 9 0 2 0 7 4 8 9 3 5 4 2 9 4 7
     7 6 4 4 2 2 1 7 5 1 0 6 3 9 6 3 2 6 0 3 0 2 6 0 6 1 2 2 1 3 9 4 0 5 3 7
     8 7 8 8 9 2 8 1 9 1 2 4 2 3 4 4 8 1 6 6 5 9 6 0 1 0 6 7 2 8 8 7 5 9 3 2
     ------------------------------------------------------------------------
     7 1 0 6 4 1 1 0 0 9 4 3 6 7 6 4 5 6 3 6 4 5 4 1 0 9 7 5 1 4 5 2 3 3 1 1
     8 3 4 6 2 2 7 3 9 9 4 7 1 8 5 5 3 1 2 5 1 1 3 3 9 1 4 3 5 7 9 2 2 6 3 1
     3 8 6 6 4 0 1 6 0 2 9 3 1 3 3 4 4 4 3 9 7 1 7 2 1 8 5 1 4 8 3 8 6 5 9 7
     0 6 9 2 9 9 0 4 5 6 4 3 0 7 4 2 4 3 6 8 9 7 3 6 1 4 1 9 0 1 3 8 8 4 5 8
     4 9 7 9 6 1 6 0 1 3 4 4 7 9 0 3 1 4 2 5 3 9 9 1 2 0 3 8 4 7 6 2 8 9 9 5
     ------------------------------------------------------------------------
     5 9 7 4 0 5 3 |
     4 9 2 2 8 3 0 |
     0 1 1 5 4 0 4 |
     0 5 8 3 3 4 8 |
     4 1 6 8 1 0 6 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 5.99668 seconds
i8 : time C = points(M,R);
     -- used 0.483928 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :