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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 2 3 2 5 |
     | 2 1 6 7 5 |
     | 8 1 3 3 5 |

              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          37 2   80 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  + --x
                                                                  13     13 
     ------------------------------------------------------------------------
       41    401    150         9 2    9    30    152    105   2   23 2  
     + --y - ---z + ---, x*z + --z  - --x + --y - ---z + ---, y  + --z  +
       13     13     13        13     13    13     13     13       13    
     ------------------------------------------------------------------------
     120    49    257    30         3 2   81    29     3    165   2   24 2  
     ---x - --y - ---z + --, x*y + --z  - --x - --y - --z + ---, x  + --z  -
      13    13     13    13        13     13    13    13     13       13    
     ------------------------------------------------------------------------
     11    54    258    150   3   292 2   280    280    1839    720
     --x + --y - ---z + ---, z  - ---z  - ---x - ---y + ----z - ---})
     13    13     13     13        13      13     13     13      13

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.41693 seconds
i8 : time C = points(M,R);
     -- used 0.474308 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :