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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 = | 0 1 4 7 1 |
     | 0 4 6 0 5 |
     | 3 1 9 9 0 |

              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          1 2   9   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + -z  + -x -
                                                                  2     2   
     ------------------------------------------------------------------------
     27    45    117        1 2   27    9    3    9   2   11 2   17    31   
     --y - --z + ---, x*z - -z  - --x + -y + -z + -, y  - --z  + --x - --y +
      4     4     4         4      4    8    8    8       12      4     8   
     ------------------------------------------------------------------------
     145    79        1 2   21    43    25    93   2   11 2   107    69   
     ---z - --, x*y - -z  + --x - --y - --z + --, x  - --z  - ---x + --y +
      24     8        8      8    16    16    16       48      16    32   
     ------------------------------------------------------------------------
     229    163   3   17 2   27    27    3    189
     ---z - ---, z  - --z  - --x - --y + -z + ---})
      96     32        2      2     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 6 8 7 9 4 9 5 3 5 3 2 2 6 1 6 0 5 4 4 0 9 8 8 9 8 3 3 9 4 3 9 2 3 4
     | 5 3 5 4 1 8 5 4 3 8 2 8 1 9 2 4 4 0 2 1 9 9 5 0 5 7 4 3 4 8 8 6 6 2 7
     | 1 8 1 3 8 2 9 7 5 8 8 1 6 0 4 7 5 4 0 1 2 2 9 3 6 8 9 5 7 1 9 8 8 9 3
     | 3 8 6 9 8 7 2 8 4 1 5 6 4 0 6 9 1 8 7 4 4 9 9 2 0 8 7 3 6 2 9 1 5 6 9
     | 7 1 3 2 5 5 7 9 3 0 6 2 4 2 5 7 2 2 5 3 1 3 4 8 7 7 2 3 2 3 1 6 0 8 8
     ------------------------------------------------------------------------
     6 7 6 1 8 5 9 3 4 5 1 5 2 0 7 5 7 4 7 0 5 1 4 0 1 3 4 9 6 4 0 9 7 9 4 3
     9 8 7 2 8 2 1 9 2 9 0 8 9 0 3 8 3 7 4 0 1 0 9 7 9 9 0 4 8 1 2 9 3 9 4 2
     8 5 9 7 7 0 6 2 5 3 8 4 8 8 4 5 8 5 7 5 3 3 6 3 6 3 1 7 8 0 8 5 4 8 9 3
     7 8 4 9 6 2 3 2 5 8 3 8 2 4 1 8 5 4 2 2 8 3 4 3 4 6 1 2 5 3 4 2 9 7 3 7
     7 9 6 3 2 9 5 5 3 3 2 5 9 5 1 1 6 7 5 3 5 4 1 8 6 4 9 8 8 9 1 6 4 9 9 1
     ------------------------------------------------------------------------
     7 0 3 6 5 2 3 7 5 5 0 6 6 1 2 7 6 9 8 3 4 0 2 4 5 0 9 7 6 4 1 0 9 4 0 1
     6 1 7 5 0 1 2 4 8 7 1 2 0 8 7 5 6 4 1 3 8 1 8 4 6 9 4 4 3 9 3 2 8 7 5 4
     2 2 2 3 0 5 9 7 5 0 9 7 4 3 4 2 4 3 2 4 2 2 6 4 6 9 7 8 3 0 9 2 5 1 4 2
     4 2 9 6 4 8 5 1 0 0 6 3 2 8 2 5 5 4 8 5 1 1 3 8 2 6 3 7 5 3 3 9 9 8 9 3
     2 3 5 1 6 8 7 2 8 1 9 0 2 5 9 5 9 9 8 6 0 2 5 7 4 4 1 9 9 8 2 3 8 7 6 3
     ------------------------------------------------------------------------
     8 2 3 8 7 4 6 0 3 0 0 4 5 1 5 8 7 2 2 4 8 1 6 6 3 7 5 7 7 5 5 5 2 8 0 6
     7 3 9 7 9 4 0 0 5 9 3 9 8 4 0 6 0 0 1 7 8 4 5 6 5 9 2 9 1 1 4 2 4 1 7 0
     3 3 2 2 1 4 5 7 8 2 0 3 8 2 4 0 8 5 5 1 6 4 2 4 3 3 1 4 1 5 7 8 4 5 5 9
     8 7 3 6 0 3 6 9 1 5 9 0 5 6 0 5 6 6 2 2 2 0 6 4 5 0 7 7 4 3 4 5 5 3 9 2
     8 8 7 1 4 8 2 1 3 7 9 2 4 3 8 6 7 8 2 0 7 9 5 8 1 1 3 4 7 9 2 1 0 7 8 8
     ------------------------------------------------------------------------
     2 8 3 7 0 8 8 |
     2 8 7 4 9 0 1 |
     9 2 2 1 1 1 8 |
     1 0 0 6 8 3 7 |
     5 6 1 3 8 4 2 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 21.3094 seconds
i8 : time C = points(M,R);
     -- used 1.63937 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :