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

              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          20 2   144 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - ---x
                                                                  73      73 
     ------------------------------------------------------------------------
       187    175    1281        32 2   120    22    450    666   2   403 2  
     - ---y - ---z + ----, x*z + --z  - ---x - --y - ---z + ---, y  + ---z  -
        73     73     73         73      73    73     73     73        73    
     ------------------------------------------------------------------------
     252    601    1675    2406        54 2   531    320    294    2520   2  
     ---x - ---y - ----z + ----, x*y + --z  - ---x - ---y - ---z + ----, x  +
      73     73     73      73         73      73     73     73     73       
     ------------------------------------------------------------------------
      82 2   577    34    268    1162   3   496 2   108    24    843    468
     ---z  - ---x - --y - ---z + ----, z  - ---z  + ---x - --y + ---z - ---})
     219      73    73    219     73         73      73    73     73     73

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.08314 seconds
i8 : time C = points(M,R);
     -- used 0.623396 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :