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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 9 6 3 5 |
     | 3 7 4 2 4 |
     | 7 7 4 4 6 |

              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            2   12   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - 3z  + --x -
                                                                         5   
     ------------------------------------------------------------------------
     38    136    304          2   17     9    353    386   2     2   16   
     --y + ---z - ---, x*z - 4z  - --x - --y + ---z - ---, y  - 5z  + --x -
      5     5      5                5    10     10     5               5   
     ------------------------------------------------------------------------
     54    258    592          2   3    99    623    706   2     2   37   
     --y + ---z - ---, x*y - 6z  + -x - --y + ---z - ---, x  - 7z  - --x -
      5     5      5               5    10     10     5               5   
     ------------------------------------------------------------------------
     12    359    786   3      2
     --y + ---z - ---, z  - 17z  + 94z - 168})
      5     5      5

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.20021 seconds
i8 : time C = points(M,R);
     -- used 0.526792 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :