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

              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          899 2   36 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  - --x
                                                                  133     19 
     ------------------------------------------------------------------------
       163    7241    1302        670 2   11    120    4093    819   2  
     - ---y - ----z + ----, x*z - ---z  + --x + ---y + ----z - ---, y  +
        19     133     19         133     19     19     133     19      
     ------------------------------------------------------------------------
     290 2   30    177    2250    648        436 2   25    81    3502    315 
     ---z  - --x - ---y - ----z + ---, x*y - ---z  - --x - --y + ----z - ---,
     133     19     19     133     19        133     19    19     133     19 
     ------------------------------------------------------------------------
      2   601 2   175    60    4507    324   3     2
     x  - ---z  - ---x + --y + ----z - ---, z  - 8z  + 7z})
          133      19    19     133     19

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 7.93105 seconds
i8 : time C = points(M,R);
     -- used 0.571143 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :