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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 9 4 1 8 |
     | 5 7 3 3 3 |
     | 5 5 6 1 9 |

              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                   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - 5y - 3z +
                                                                           
     ------------------------------------------------------------------------
               53 2   339    44    303    102   2   22 2   240    948    10 
     15, x*z - --z  - ---x + --y + ---z - ---, y  - --z  + ---x - ---y + --z
               59      59    59     59     59       59      59     59    59 
     ------------------------------------------------------------------------
       2085        22 2   63    889    10    2439   2   67 2   315    629   
     + ----, x*y - --z  + --x - ---y + --z + ----, x  - --z  - ---x - ---y +
        59         59     59     59    59     59        59      59     59   
     ------------------------------------------------------------------------
     481    1729   3   900 2   480    480    4051    4170
     ---z + ----, z  - ---z  - ---x + ---y + ----z - ----})
      59     59         59      59     59     59      59

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 7.95198 seconds
i8 : time C = points(M,R);
     -- used 0.633431 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :