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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 = | 0 9 4 8 9 |
     | 3 7 8 8 0 |
     | 3 4 9 3 8 |

              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          131 2   45 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  + --x
                                                                   80     16 
     ------------------------------------------------------------------------
       15    1031    6087         3 2   231    30    147    495   2   551 2  
     - --y - ----z + ----, x*z + --z  - ---x + --y - ---z + ---, y  - ---z  -
        2     40      80         52      52    13     26     52       260    
     ------------------------------------------------------------------------
     105    101    3131    10107        243 2   719    189    1143    9207 
     ---x - ---y + ----z - -----, x*y + ---z  - ---x - ---y - ----z + ----,
      52     13     130     260         208     208     26     104     208 
     ------------------------------------------------------------------------
      2   119 2   114    16    1288    2553   3   783 2   75    30    1681   
     x  + ---z  - ---x + --y - ----z + ----, z  - ---z  + --x - --y + ----z -
           65      13    13     65      65         52     52    13     26    
     ------------------------------------------------------------------------
     4083
     ----})
      52

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 7.97465 seconds
i8 : time C = points(M,R);
     -- used 0.421101 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :