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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 7 7 2 |
     | 7 9 2 5 4 |
     | 0 1 5 7 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          407 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  +
                                                                  448    
     ------------------------------------------------------------------------
     675    209     89    1237        115 2   975    395    3331    6665   2
     ---x - ---y - ---z - ----, x*z + ---z  - ---x - ---y - ----z + ----, y 
     224    224    448     224        448     224    224     448     224    
     ------------------------------------------------------------------------
       977 2   1125    3385    6273    17219        1655 2    61     81   
     + ---z  - ----x - ----y - ----z + -----, x*y - ----z  - ---x - ---y +
       448      224     224     448     224          896     448    448   
     ------------------------------------------------------------------------
     10695    11733   2   15 2   129    15    135    397   3   75 2   30   
     -----z - -----, x  + --z  - ---x - --y - ---z + ---, z  - --z  + --x -
      896      448        28      14    14     28     14        7      7   
     ------------------------------------------------------------------------
     18    164    6
     --y + ---z + -})
      7     7     7

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 5.41318 seconds
i8 : time C = points(M,R);
     -- used 0.570913 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :