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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 = | 8 1 1 1 4 |
     | 2 1 6 0 0 |
     | 8 8 7 5 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          23 2    2 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - --x
                                                                  45     15 
     ------------------------------------------------------------------------
       106    19    844        56 2   119     7    46    1918   2   11 2   2 
     - ---y + --z - ---, x*z - --z  - ---x - --y + --z - ----, y  + --z  + -x
        15     3     45        45      15    15     3     45         5     5 
     ------------------------------------------------------------------------
       29          398        112 2   28    29    98    4466   2   224 2  
     - --y - 27z + ---, x*y - ---z  - --x - --y + --z - ----, x  - ---z  -
        5           5          45     15    15     3     45         45    
     ------------------------------------------------------------------------
     131    28    196    8752   3      2
     ---x - --y + ---z - ----, z  - 20z  + 131z - 280})
      15    15     3      45

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.10458 seconds
i8 : time C = points(M,R);
     -- used 0.564422 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :