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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 = | 5 9 3 2 8 |
     | 8 0 6 0 3 |
     | 7 5 5 8 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           7 2   111 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  - ---x
                                                                  61      61 
     ------------------------------------------------------------------------
       416    350    2574        66 2   332    27    1043    3808   2    53 2
     - ---y - ---z + ----, x*z + --z  - ---x - --y - ----z + ----, y  + ---z 
        61     61     61         61      61    61     61      61        122  
     ------------------------------------------------------------------------
       213    945    593    3261        259 2   285    651    1351    2235 
     - ---x - ---y - ---z + ----, x*y - ---z  - ---x - ---y + ----z - ----,
       122    122     61     61         122     122    122     61      61  
     ------------------------------------------------------------------------
      2    55 2   1273    191    277    1485   3   960 2   60    60    4751 
     x  + ---z  - ----x + ---y - ---z + ----, z  - ---z  + --x + --y + ----z
          122      122    122     61     61         61     61    61     61  
     ------------------------------------------------------------------------
       7920
     - ----})
        61

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 2.65526 seconds
i8 : time C = points(M,R);
     -- used 0.376748 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :