next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 = | 6 9 4 1 4 |
     | 9 1 8 1 9 |
     | 2 2 1 5 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          1468 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ----z  -
                                                                  1149    
     ------------------------------------------------------------------------
     920    1111    589    28213        333 2   1126    135    355    4396 
     ---x - ----y + ---z + -----, x*z - ---z  - ----x - ---y - ---z + ----,
     383     383    383     1149        766      383    383    766     383 
     ------------------------------------------------------------------------
      2    952 2   280    3935    2968    32216        2872 2   687    2412 
     y  + ----z  - ---x - ----y - ----z + -----, x*y + ----z  - ---x - ----y
          1149     383     383     383     1149        1149     383     383 
     ------------------------------------------------------------------------
       7512    49028   2    85 2   3905    690    795    4682   3   4274 2  
     - ----z + -----, x  + ---z  - ----x + ---y - ---z + ----, z  - ----z  -
        383     1149       383      383    383    383     383        383    
     ------------------------------------------------------------------------
     960    360    12421    1810
     ---x - ---y + -----z - ----})
     383    383     383      383

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.59381 seconds
i8 : time C = points(M,R);
     -- used 0.531933 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :