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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 = | 5 1 9 0 7 |
     | 5 3 7 6 4 |
     | 3 4 8 5 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           91 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  +
                                                                  139    
     ------------------------------------------------------------------------
      36    569     60    1219        182 2   623    252    815    1037   2  
     ---x - ---y + ---z + ----, x*z - ---z  - ---x + ---y + ---z - ----, y  -
     139    139    139     139        139     139    139    139     139      
     ------------------------------------------------------------------------
      67 2    54    1201    507    1342        302 2   574     63    2750   
     ---z  + ---x - ----y + ---z + ----, x*y - ---z  - ---x - ---y + ----z -
     139     139     139    139     139        139     139    139     139   
     ------------------------------------------------------------------------
     5822   2    16 2   985     42    576    3532   3   2283 2    60    210 
     ----, x  - ---z  - ---x - ---y - ---z + ----, z  - ----z  + ---x + ---y
      139       139     139    139    139     139        139     139    139 
     ------------------------------------------------------------------------
       11498    19050
     + -----z - -----})
        139      139

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 8.39902 seconds
i8 : time C = points(M,R);
     -- used 0.589305 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :