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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 = | 0 0 0 3 2 |
     | 8 2 9 1 5 |
     | 1 7 0 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           2   7   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + z  - -x -
                                                                       4   
     ------------------------------------------------------------------------
     15    51    135        29    21    21    189   2    2   11    37    35 
     --y - --z + ---, x*z - --x - --y - --z + ---, y  - z  + --x - --y + --z
      4     4     4          4     4     4     4              4     4     4 
     ------------------------------------------------------------------------
       9                            2   11    3    3    27   3     2   139   
     + -, x*y - 2x + 3y + 3z - 27, x  - --x - -y - -z + --, z  - 8z  - ---x -
       4                                 4    4    4     4              4    
     ------------------------------------------------------------------------
     159    131    1431
     ---y - ---z + ----})
      4      4       4

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 9.984 seconds
i8 : time C = points(M,R);
     -- used 0.874 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :