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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 = | 4 1 9 0 9 |
     | 2 0 7 8 9 |
     | 5 3 5 4 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          863 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  -
                                                                  284    
     ------------------------------------------------------------------------
     165    545    6739              759 2   735     25    6331         2  
     ---x - ---y - ----z + 45, x*z + ---z  - ---x + ---y - ----z + 45, y  +
     142    142     284              284     142    142     284            
     ------------------------------------------------------------------------
     100 2   18    621    294         411 2   627    154    1667         2  
     ---z  - --x - ---y - ---z, x*y + ---z  - ---x - ---y - ----z + 18, x  +
      71     71     71     71         142      71     71     142            
     ------------------------------------------------------------------------
     279 2   773    150    1455         3   567 2   30    30    1072
     ---z  - ---x - ---y - ----z + 36, z  - ---z  - --x + --y + ----z})
      71      71     71     71               71     71    71     71

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.40006 seconds
i8 : time C = points(M,R);
     -- used 0.449691 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :