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 = | 8 2 0 5 8 |
     | 1 3 8 3 4 |
     | 2 2 3 7 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           379 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ----z  -
                                                                  1072    
     ------------------------------------------------------------------------
     155    1537    6441    1847         117 2   837    705    6695    969 
     ---x - ----y - ----z + ----, x*z + ----z  - ---x + ---y - ----z + ---,
     536     536    1072     134        1072     536    536    1072    134 
     ------------------------------------------------------------------------
      2   1599 2   1255    5909    12885    6275        255 2   26    11   
     y  + ----z  - ----x - ----y - -----z + ----, x*y - ---z  - --x - --y +
          1072      536     536     1072     134        134     67    67   
     ------------------------------------------------------------------------
     2085    1892   2   1131 2   5947    1761    7545    5079   3   2565 2  
     ----z - ----, x  + ----z  - ----x - ----y - ----z + ----, z  - ----z  +
      134     67        1072      536     536    1072     134        268    
     ------------------------------------------------------------------------
     105    315    5003    1050
     ---x + ---y + ----z - ----})
     134    134     268     67

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 9.30388 seconds
i8 : time C = points(M,R);
     -- used 0.562629 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :