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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 = | 2 6 4 5 8 |
     | 3 5 8 7 4 |
     | 5 5 6 5 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          4 2       
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - -z  - 5y +
                                                                  3         
     ------------------------------------------------------------------------
     20         4 2        56         2   49 2   8    56    473    83       
     --z, x*z + -z  - 5x - --z + 60, y  - --z  + -x - --y + ---z - --, x*y +
      3         3           3             30     5     5     30     5       
     ------------------------------------------------------------------------
     7 2   19    22    59    194   2   28 2   43    6    296    212   3  
     -z  - --x - --y - --z + ---, x  - --z  - --x + -y + ---z - ---, z  -
     5      5     5     5     5        15      5    5     15     5
     ------------------------------------------------------------------------
        2
     14z  + 63z - 90})

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 7.89105 seconds
i8 : time C = points(M,R);
     -- used 0.454933 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :