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

              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          671 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  -
                                                                  625    
     ------------------------------------------------------------------------
     438    3276    9328    30318        361 2   2867    791    623    3537 
     ---x - ----y - ----z + -----, x*z - ---z  - ----x + ---y + ---z + ----,
     625     625     625     625         625      625    625    625     625 
     ------------------------------------------------------------------------
      2    32 2    54    933    326    844        202 2   1419    2163   
     y  - ---z  - ---x - ---y + ---z + ---, x*y - ---z  - ----x - ----y +
          125     125    125    125    125        625      625     625   
     ------------------------------------------------------------------------
     3136    4141   2   134 2   523    304    1412    4622   3   10107 2  
     ----z - ----, x  - ---z  - ---x + ---y + ----z - ----, z  - -----z  -
      625     625       125     125    125     125     125        625     
     ------------------------------------------------------------------------
     204    2892    48926    78756
     ---x + ----y + -----z - -----})
     625     625     625      625

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 7.98228 seconds
i8 : time C = points(M,R);
     -- used 0.477388 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :