next | previous | forward | backward | up | top | index | toc | Macaulay2 web site

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

              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          1 2      
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - -z  - y +
                                                                  2        
     ------------------------------------------------------------------------
     3             7 2       49    21   2   53 2              379    476     
     -z - 1, x*z + -z  - x - --z + --, y  + --z  + 2x - 13y - ---z + ---, x*y
     2             4          4     2        5                 5      5      
     ------------------------------------------------------------------------
       121 2        9    733    69   2   19 2   19    3    58    369   3  
     - ---z  - 8x - -y + ---z + --, x  + --z  - --x - -y - --z + ---, z  -
        20          2     20    10       20      2    4     5     10
     ------------------------------------------------------------------------
       2
     9z  + 20z - 12})

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 7.20845 seconds
i8 : time C = points(M,R);
     -- used 0.636039 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :