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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 = | 7 7 7 3 9 |
     | 1 8 2 1 3 |
     | 6 4 2 6 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          43 2   49 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - --y
                                                                  88     11 
     ------------------------------------------------------------------------
       101    25        13 2         2    103    428   2   23 2   97    108 
     + ---z + --, x*z - --z  - 6x - --y - ---z + ---, y  + --z  - --y - ---z
        44    11        44          11     22     11       22     11     11 
     ------------------------------------------------------------------------
       320        13 2       79    51    43   2   39 2          6    153   
     + ---, x*y - --z  - x - --y + --z + --, x  - --z  - 10x - --y + ---z +
        11        44         11    22    11       44           11     22   
     ------------------------------------------------------------------------
     129   3   171 2   24    790    936
     ---, z  - ---z  - --y + ---z - ---})
      11        11     11     11     11

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.47466 seconds
i8 : time C = points(M,R);
     -- used 0.495161 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :