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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 = | 6 4 5 4 5 |
     | 0 8 6 5 9 |
     | 6 0 6 4 1 |

              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           29 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  +
                                                                  237    
     ------------------------------------------------------------------------
     420    404    2210    1552         85 2   234    40    518    616   2  
     ---x - ---y - ----z + ----, x*z - ---z  - ---x + --y - ---z + ---, y  +
      79     79     237     79         237      79    79    237     79      
     ------------------------------------------------------------------------
     12 2   966    635    246    3888        23 2   390    460    200   
     --z  - ---x - ---y + ---z + ----, x*y + --z  - ---x - ---y - ---z +
     79      79     79     79     79         79      79     79     79   
     ------------------------------------------------------------------------
     2712   2    1 2   749    20    19    1572   3   705 2   720    120   
     ----, x  - --z  - ---x + --y + --z + ----, z  - ---z  - ---x - ---y +
      79        79      79    79    79     79         79      79     79   
     ------------------------------------------------------------------------
     1466    3840
     ----z + ----})
      79      79

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.86 seconds
i8 : time C = points(M,R);
     -- used 1.2 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :