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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 = | 7 4 4 7 8 |
     | 5 9 0 5 6 |
     | 4 1 2 0 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           19 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  -
                                                                  115    
     ------------------------------------------------------------------------
     107    181    651    1654         33 2   204     42    937    1638   2  
     ---x - ---y - ---z + ----, x*z + ---z  - ---x - ---y - ---z + ----, y  +
     115    115    115     115        115     115    115    115     115      
     ------------------------------------------------------------------------
     234 2   498    1061    936    1056        108 2   699    472    432   
     ---z  + ---x - ----y - ---z - ----, x*y + ---z  - ---x - ---y - ---z +
     115     115     115    115     115        115     115    115    115   
     ------------------------------------------------------------------------
     3228   2   108 2   1389     12    432    4148   3   153 2   30     6   
     ----, x  + ---z  - ----x - ---y - ---z + ----, z  - ---z  + --x - --y +
      115       115      115    115    115     115        23     23    23   
     ------------------------------------------------------------------------
     244    180
     ---z - ---})
      23     23

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.2002 seconds
i8 : time C = points(M,R);
     -- used 0.406543 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :