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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 5 2 1 3 |
     | 0 5 9 1 7 |
     | 2 1 8 0 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          80 2   144 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + ---x
                                                                  83     415 
     ------------------------------------------------------------------------
       566     13    422        36 2   264    346    2583     82   2   95 2  
     - ---y + ---z + ---, x*z + --z  - ---x + ---y - ----z - ---, y  - --z  -
       415    415    415        83     415    415     415    415       83    
     ------------------------------------------------------------------------
     576    2301    2023    2462        45 2   247    2182    469    2014   2
     ---x - ----y + ----z + ----, x*y + --z  - ---x - ----y - ---z + ----, x 
     415     415     415     415        83     415     415    415     415    
     ------------------------------------------------------------------------
       37 2   737    212    145    442   3      2   48    8    294    56
     - --z  - ---x + ---y + ---z + ---, z  - 15z  - --x - -y + ---z + --})
       83      83     83     83     83               5    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 = | 6 2 7 4 6 7 8 2 0 3 5 7 5 9 2 6 8 2 1 1 7 6 9 6 6 6 6 0 6 6 1 6 0 5 6
     | 3 1 5 3 8 0 9 3 1 0 9 4 9 6 4 2 5 8 5 5 2 3 6 6 1 3 9 0 5 4 2 4 2 6 9
     | 6 4 9 5 7 9 5 8 0 3 0 4 6 0 9 3 9 6 1 0 6 1 0 2 2 5 5 6 2 8 8 0 9 6 0
     | 2 6 3 3 1 7 2 6 0 8 0 0 1 7 7 5 9 4 4 4 6 7 1 9 8 8 6 9 0 2 1 0 9 4 8
     | 2 7 4 5 3 8 9 1 4 2 9 0 1 9 9 7 1 4 3 3 2 7 2 7 5 6 4 5 3 2 5 9 7 9 4
     ------------------------------------------------------------------------
     4 6 4 0 3 8 1 5 7 1 6 0 9 7 7 6 3 5 9 5 1 1 7 8 5 0 9 1 5 6 7 6 4 2 3 2
     9 4 4 0 0 3 8 3 0 7 5 1 5 1 1 0 0 5 5 7 7 3 7 6 3 0 7 2 9 9 9 8 4 3 6 8
     5 9 9 3 6 5 3 5 6 7 0 9 7 8 1 8 0 1 0 5 6 3 9 3 0 5 5 6 5 1 4 3 7 3 2 2
     0 1 1 0 4 4 6 3 6 7 5 9 4 1 9 4 3 5 0 6 4 6 3 8 7 5 7 1 4 9 1 7 1 4 1 6
     4 8 2 4 8 4 7 0 2 5 9 8 4 4 1 0 9 9 7 8 1 7 1 9 2 0 0 6 2 0 1 5 6 7 5 9
     ------------------------------------------------------------------------
     3 7 4 8 9 0 3 4 2 1 2 4 7 3 3 6 9 3 5 6 6 2 4 1 0 9 7 6 2 4 3 9 4 2 7 9
     2 5 2 8 4 9 3 9 1 1 2 0 6 2 8 8 3 1 5 9 4 5 7 7 5 9 7 2 7 1 2 3 4 5 3 1
     5 6 5 3 1 0 0 4 2 8 6 6 8 4 1 6 7 6 8 8 3 9 4 9 7 6 1 3 4 4 9 5 8 9 4 2
     6 3 7 2 5 5 3 1 8 6 6 4 4 2 1 6 0 7 4 8 5 9 6 8 7 3 4 1 3 3 4 6 7 1 2 1
     7 0 5 6 1 0 5 1 1 2 7 9 0 5 5 7 3 4 3 7 0 5 3 5 2 0 2 1 0 0 0 8 6 5 5 1
     ------------------------------------------------------------------------
     2 4 5 7 2 7 8 9 7 6 4 5 6 3 6 6 2 0 2 5 8 6 7 9 1 9 2 6 9 3 2 2 4 3 6 3
     4 0 0 7 1 5 5 1 6 1 9 4 2 2 6 1 1 1 1 4 2 2 3 9 2 6 5 4 2 4 1 7 5 9 4 8
     1 3 0 8 4 6 9 8 8 9 5 1 7 9 8 1 4 2 9 5 3 9 1 7 7 9 3 2 1 9 8 5 1 8 8 6
     0 7 4 6 8 9 7 9 9 0 5 6 3 6 8 4 1 7 6 8 2 5 2 8 6 4 1 7 5 8 2 8 6 5 0 8
     1 5 5 8 7 3 1 5 9 2 1 3 8 9 4 8 0 6 2 5 4 9 2 8 9 5 9 7 8 2 9 7 4 0 0 9
     ------------------------------------------------------------------------
     6 6 8 7 5 4 1 |
     3 7 6 2 8 1 6 |
     4 9 5 2 5 2 2 |
     0 5 8 4 6 8 8 |
     8 0 6 4 5 3 0 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 9.40399 seconds
i8 : time C = points(M,R);
     -- used 0.772349 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :