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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 2 5 9 |
     | 4 0 9 4 5 |
     | 0 7 9 8 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          437 2   36 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  - --x
                                                                   26     13 
     ------------------------------------------------------------------------
       252    263              229 2   36    252    133         2   5660 2  
     + ---y + ---z - 72, x*z - ---z  - --x + ---y + ---z - 72, y  - ----z  -
        13     2                26     13     13     2               91     
     ------------------------------------------------------------------------
     1280    1163    3520    2420        1295 2   88    1097    805        
     ----x + ----y + ----z - ----, x*y + ----z  + --x - ----y - ---z + 316,
      91      13       7       7          26      13     13      2         
     ------------------------------------------------------------------------
      2   480 2   251    756                 3   528 2   72    504
     x  - ---z  - ---x + ---y + 300z - 198, z  - ---z  - --x + ---y + 263z -
           13      13     13                      13     13     13
     ------------------------------------------------------------------------
     144})

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.119 seconds
i8 : time C = points(M,R);
     -- used 0.555808 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :