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

              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          18 2   16 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  + --x
                                                                  35     35 
     ------------------------------------------------------------------------
       232    363    1544         1 2   312    114    114    762   2    4 2  
     - ---y - ---z + ----, x*z - --z  - ---x + ---y - ---z + ---, y  - --z  -
        35     35     35         35      35     35     35     35       15    
     ------------------------------------------------------------------------
      8    124    44    173         1  2   347    691     79    2897   2  
     --x - ---y + --z + ---, x*y - ---z  - ---x - ---y - ---z + ----, x  +
     15     15    15     15        105     105    105    105     105      
     ------------------------------------------------------------------------
      11 2   593    694    391    5198   3   401 2   608     4    1046   
     ---z  - ---x - ---y - ---z + ----, z  - ---z  + ---x + --y + ----z -
     105     105    105    105     105        35      35    35     35    
     ------------------------------------------------------------------------
     3068
     ----})
      35

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.39094 seconds
i8 : time C = points(M,R);
     -- used 1.09021 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :