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

              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          175 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  +
                                                                  481    
     ------------------------------------------------------------------------
     280    3202    2098    20576        1115 2   1102    1120    7302   
     ---x - ----y - ----z + -----, x*z - ----z  - ----x + ----y + ----z -
     481     481     481     481          481      481     481     481   
     ------------------------------------------------------------------------
     17984   2   210 2   336    4527    2100    11480        315 2   3344   
     -----, y  + ---z  - ---x - ----y - ----z + -----, x*y - ---z  - ----x -
      481        481     481     481     481     481         481      481   
     ------------------------------------------------------------------------
     4032    3150    23184   2   810 2   3995    1176    8100    21192   3  
     ----y + ----z + -----, x  - ---z  - ----x + ----y + ----z - -----, z  -
      481     481     481        481      481     481     481     481       
     ------------------------------------------------------------------------
     7721 2    40    320    36806    44336
     ----z  + ---x - ---y + -----z - -----})
      481     481    481     481      481

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 20.9237 seconds
i8 : time C = points(M,R);
     -- used 1.84084 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :