points -- produces the ideal and initial ideal from the coordinates of a finite set of points

Synopsis

Usage:
(Q,inG,G) = points(M,R)
Inputs:
- M, a matrix, in which each column consists of the coordinates of a point
- R, a polynomial ring, coordinate ring of the affine space containing the points
Outputs:
- Q, a list, list of standard monomials
- inG, an ideal, initial ideal of the set of points
- G, a list, list of generators for Grobner basis for ideal of points

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 = | 4 8 1 7 4 |
     | 6 1 4 2 1 |
     | 2 8 9 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          329 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  +
                                                                  263    
     ------------------------------------------------------------------------
     672     22    2999    10394        461 2   2542    1512    6630   
     ---x - ---y + ----z - -----, x*z + ---z  - ----x - ----y - ----z +
     263    263     263     263         263      263     263     263   
     ------------------------------------------------------------------------
     28552   2   186 2   268    2305    2068    8826         72 2   494   
     -----, y  + ---z  - ---x - ----y - ----z + ----, x*y + ---z  - ---x -
      263        263     263     263     263     263        263     263   
     ------------------------------------------------------------------------
     1028    546    2636   2   260 2   2541    264    2410    592   3  
     ----y - ---z + ----, x  - ---z  - ----x + ---y + ----z + ---, z  -
      263    263     263       263      263    263     263    263      
     ------------------------------------------------------------------------
     4537 2   672    504    23250    36168
     ----z  + ---x + ---y + -----z - -----})
      263     263    263     263      263

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 9.22567 seconds

i8 : time C = points(M,R);
     -- used 0.607809 seconds

i9 : J == C_2  

o9 = true

Ways to use `points` :

points(Matrix,Ring)

points -- produces the ideal and initial ideal from the coordinates of a finite set of points

Synopsis

Description

See also

Ways to use points :

Ways to use `points` :