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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 = | 9 3 0 0 4 |
     | 5 9 8 1 8 |
     | 9 6 0 6 1 |

              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           57 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  +
                                                                  226    
     ------------------------------------------------------------------------
      40    693    2185    5544        391 2   318    135    2031    1080   2
     ---x - ---y - ----z + ----, x*z - ---z  - ---x - ---y + ----z + ----, y 
     113    113     226     113        226     113    113     226     113    
     ------------------------------------------------------------------------
       166 2   344    1173    514    2152         92 2   761     96    664   
     + ---z  + ---x - ----y - ---z + ----, x*y + ---z  - ---x - ---y - ---z +
       339     339     113    113     113        113     113    113    113   
     ------------------------------------------------------------------------
     768   2   149 2   675    126    1041    1008   3   1475 2   960    360 
     ---, x  - ---z  - ---x + ---y + ----z - ----, z  - ----z  - ---x + ---y
     113       113     113    113     113     113        113     113    113 
     ------------------------------------------------------------------------
       5202    2880
     + ----z - ----})
        113     113

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 21.8953 seconds
i8 : time C = points(M,R);
     -- used 1.82803 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :