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

              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          188 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  +
                                                                  171    
     ------------------------------------------------------------------------
     104    778    2615    8194        16 2   338     8    265    986   2  
     ---x - ---y - ----z + ----, x*z + --z  - ---x - --y - ---z + ---, y  +
     171    171     171     171        57      57    57     57     57      
     ------------------------------------------------------------------------
      2 2   28    191    26    280        12 2    3    13    156    483   2  
     --z  - --x - ---y - --z + ---, x*y - --z  - --x - --y + ---z - ---, x  +
     19     19     19    19     19        19     19    19     19     19      
     ------------------------------------------------------------------------
     20 2   451    10    260    1204   3   2483 2   220    212    10220   
     --z  - ---x - --y - ---z + ----, z  - ----z  + ---x - ---y + -----z -
     57      57    57     57     57         171     171    171     171    
     ------------------------------------------------------------------------
     10408
     -----})
      171

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.62646 seconds
i8 : time C = points(M,R);
     -- used 0.56248 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :