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

              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           9 2   24 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  - --x
                                                                  13     13 
     ------------------------------------------------------------------------
       77    50    191        161 2   109    22    2713    1986   2   188 2  
     - --y + --z + ---, x*z + ---z  - ---x - --y - ----z + ----, y  + ---z  -
       13    13     13         78      13    13     78      13         39    
     ------------------------------------------------------------------------
     38    149    2512    3170        1 2             9         2   77 2  
     --x - ---y - ----z + ----, x*y - -z  - 4x - 6y + -z + 18, x  - --z  -
     13     13     39      13         2               2             39    
     ------------------------------------------------------------------------
     145    12    1033    776   3   243 2   24    12    1454    2604
     ---x + --y + ----z - ---, z  - ---z  - --x - --y + ----z - ----})
      13    13     39      13        13     13    13     13      13

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 6.86977 seconds
i8 : time C = points(M,R);
     -- used 0.499221 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :