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 = | 8 5 4 7 9 |
     | 3 0 1 2 1 |
     | 5 6 7 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           2 2    7 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - --x
                                                                  11     11 
     ------------------------------------------------------------------------
       56     2    119         71 2   1407    294    2250    12369   2  
     - --y - --z + ---, x*z + ---z  - ----x + ---y - ----z + -----, y  +
       11    11     11        187      187    187     187     187       
     ------------------------------------------------------------------------
      47 2    28    477    349    542         47 2   215    1225    349   
     ---z  - ---x - ---y - ---z + ---, x*y + ---z  - ---x - ----y - ---z +
     187     187    187    187    187        187     187     187    187   
     ------------------------------------------------------------------------
     1477   2    28 2   1913    432    566    2502   3   2316 2   210    630 
     ----, x  - ---z  - ----x - ---y + ---z + ----, z  - ----z  + ---x - ---y
      187       187      187    187    187     187        187     187    187 
     ------------------------------------------------------------------------
       7199    1260
     + ----z - ----})
        187     187

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.55031 seconds
i8 : time C = points(M,R);
     -- used 0.52892 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :