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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 1 5 0 |
     | 3 6 0 0 8 |
     | 6 1 0 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          21 2   30 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + --x
                                                                  11     11 
     ------------------------------------------------------------------------
       36    81    30        389 2   505     32    1084    505   2   277 2  
     - --y + --z - --, x*z - ---z  - ---x - ---y + ----z + ---, y  + ---z  +
       11    11    11        253     253    253     253    253       253    
     ------------------------------------------------------------------------
     610    1557    1873    610        269 2   1250    733    2345    1250 
     ---x - ----y - ----z - ---, x*y - ---z  - ----x - ---y + ----z + ----,
     253     253     253    253        253      253    253     253     253 
     ------------------------------------------------------------------------
      2   432 2   1113    310    1836    860   3   2462 2   380    490   
     x  - ---z  - ----x - ---y + ----z + ---, z  - ----z  - ---x - ---y +
          253      253    253     253    253        253     253    253   
     ------------------------------------------------------------------------
     6289    380
     ----z + ---})
      253    253

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 3.18552 seconds
i8 : time C = points(M,R);
     -- used 0.766883 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :