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

              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          1761 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ----z  +
                                                                   686    
     ------------------------------------------------------------------------
     180    286    18591    22824        1867 2   1672    150    18407   
     ---x - ---y + -----z - -----, x*z - ----z  - ----x + ---y + -----z -
     343     49     686      343          686      343     49     686    
     ------------------------------------------------------------------------
     22328   2   2160 2   186    103    23820    59970        1587 2   993   
     -----, y  - ----z  - ---x - ---y + -----z - -----, x*y - ----z  - ---x -
      343         343     343     49     343      343          686     343   
     ------------------------------------------------------------------------
     116    19359    23910   2   983 2   3413    180    10221    31870   3  
     ---y + -----z - -----, x  + ---z  - ----x - ---y - -----z + -----, z  -
      49     686      343        343      343     49     343      343       
     ------------------------------------------------------------------------
     7302 2   120    60    48683    100032
     ----z  - ---x + --y + -----z - ------})
      343     343    49     343       343

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 3.84818 seconds
i8 : time C = points(M,R);
     -- used 0.413048 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :