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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 = | 4 2 5 2 7 |
     | 1 7 9 3 3 |
     | 5 6 6 9 2 |

              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           19 2    7 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  - --x
                                                                  144     36 
     ------------------------------------------------------------------------
       137    661    761        37 2   73    1    643    623   2   7 2   14 
     - ---y - ---z + ---, x*z + --z  - --x + -y - ---z + ---, y  + -z  - --x
        24    144     36        48     12    8     48     12       6      3 
     ------------------------------------------------------------------------
            97    235         43 2   283    89    973    2069   2    7 2  
     - 9y - --z + ---, x*y + ---z  - ---x - --y - ---z + ----, x  - --z  -
             6     3         144      36    24    144     36        12    
     ------------------------------------------------------------------------
     20    1    97    44   3   389 2   7    7    1787    535
     --x - -y + --z - --, z  - ---z  - -x + -y + ----z - ---})
      3    2    12     3        24     6    4     24      6

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 21.1027 seconds
i8 : time C = points(M,R);
     -- used 1.57725 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :