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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 = | 9 8 0 9 6 |
     | 6 0 6 5 2 |
     | 1 3 4 8 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          793 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  -
                                                                  178    
     ------------------------------------------------------------------------
     535    845    8243    13076        579 2   762    420    6933    11754 
     ---x - ---y - ----z + -----, x*z + ---z  - ---x - ---y - ----z + -----,
      89     89     178      89         178      89     89     178      89  
     ------------------------------------------------------------------------
      2   309 2   195    748    2715    4242        519 2   165    318   
     y  + ---z  - ---x - ---y - ----z + ----, x*y - ---z  - ---x - ---y +
          178      89     89     178     89         178      89     89   
     ------------------------------------------------------------------------
     4809    3558   2   813 2   1378    735    7527    12960   3   440 2  
     ----z - ----, x  + ---z  - ----x - ---y - ----z + -----, z  - ---z  -
      178     89        178      89      89     178      89         89    
     ------------------------------------------------------------------------
     1000    924    2669    17564
     ----x - ---y - ----z + -----})
      89      89     89       89

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.15577 seconds
i8 : time C = points(M,R);
     -- used 0.329376 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :