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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 = | 0 7 8 6 5 |
     | 6 9 9 9 4 |
     | 5 4 6 0 7 |

              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           15 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  +
                                                                  317    
     ------------------------------------------------------------------------
     360    2480    2883    20160        203 2   1368    1123    1471   
     ---x - ----y - ----z + -----, x*z - ---z  - ----x - ----y - ----z +
     317     317     317     317         634      317     634     317   
     ------------------------------------------------------------------------
     26523   2    15 2   360    3860     30    11223         75 2   1953   
     -----, y  + ---z  - ---x - ----y + ---z + -----, x*y - ---z  - ----x -
      634        317     317     317    317     317         634      317   
     ------------------------------------------------------------------------
     4475     75    29475   2   158 2   1597    1925     1     15495   3  
     ----y - ---z + -----, x  - ---z  - ----x - ----y + ---z + -----, z  -
      634    317     634        317      317     317    317     317       
     ------------------------------------------------------------------------
     3126 2   1056    2097    7696    12537
     ----z  - ----x + ----y + ----z - -----})
      317      317     317     317     317

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.22494 seconds
i8 : time C = points(M,R);
     -- used 0.570397 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :