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

              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          43 2   30 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  - --x
                                                                  39     13 
     ------------------------------------------------------------------------
       37    709              59 2   72    10    815         2   1 2        
     - --y - ---z + 62, x*z + --z  - --x + --y - ---z + 64, y  - -z  - 26x +
       13     39              39     13    13     39             3          
     ------------------------------------------------------------------------
          47               43 2   230    50    85         2   25 2   224   
     9y - --z + 146, x*y - --z  - ---x + --y + --z + 62, x  - --z  - ---x +
           3               39      13    13    39             13      13   
     ------------------------------------------------------------------------
     60    190         3   229 2   180    90    1392
     --y + ---z + 15, z  - ---z  + ---x - --y + ----z - 252})
     13     13              13      13    13     13

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 12.4283 seconds
i8 : time C = points(M,R);
     -- used 1.07573 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :