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

              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          1145 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ----z  -
                                                                  1482    
     ------------------------------------------------------------------------
     811    413    1297    3657         85 2   1613    141    1815    6311 
     ---x - ---y + ----z + ----, x*z - ---z  - ----x + ---y - ----z + ----,
     247    247    1482     247        494      247    247     494     247 
     ------------------------------------------------------------------------
      2    88 2   412    2579    2102    3980         83 2   1206    1138   
     y  - ---z  - ---x - ----y + ----z + ----, x*y - ---z  - ----x - ----y +
          741     247     247     741     247        494      247     247   
     ------------------------------------------------------------------------
     349    4974   2   122 2   2143     22    388    4642   3   2420 2  
     ---z + ----, x  - ---z  - ----x - ---y + ---z + ----, z  - ----z  -
     494     247       741      247    247    741     247        247    
     ------------------------------------------------------------------------
     4350    270    4947    17130
     ----x + ---y + ----z + -----})
      247    247     247     247

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 5.01524 seconds
i8 : time C = points(M,R);
     -- used 0.538918 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :