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

              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          68 2   30 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  - --x
                                                                  13     13 
     ------------------------------------------------------------------------
       108    540    1062        60 2    2    80    361    726   2   11 2  
     - ---y - ---z + ----, x*z - --z  + --x + --y + ---z - ---, y  + --z  +
        13     13     13         13     13    13     13     13        6    
     ------------------------------------------------------------------------
     7    38    19              365 2   112    122    801    605   2   509 2
     -x - --y - --z + 26, x*y - ---z  - ---x + ---y + ---z - ---, x  + ---z 
     3     3     2               78      39     39     26     13        78  
     ------------------------------------------------------------------------
       452    335    1121    1340   3   89 2   20    20    56    240
     - ---x - ---y - ----z + ----, z  - --z  - --x - --y + --z + ---})
        39     39     26      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 = | 8 6 6 3 8 3 4 3 0 4 4 9 1 1 4 3 4 3 8 3 0 5 8 7 6 5 4 8 2 7 3 2 6 5 1
     | 1 3 6 0 6 9 9 4 8 4 4 0 3 3 7 4 5 3 5 0 1 7 9 9 3 5 3 8 9 9 8 0 2 1 5
     | 1 5 3 3 2 9 8 5 3 1 2 2 5 6 5 0 4 7 3 3 4 2 4 5 0 0 5 2 8 3 2 6 6 7 7
     | 9 8 4 2 0 7 8 6 6 6 7 1 5 6 5 0 2 6 1 5 7 5 7 3 1 1 0 9 5 6 9 6 9 3 1
     | 9 8 0 1 2 9 3 1 4 7 5 4 5 9 6 8 3 1 4 6 8 8 5 4 5 9 4 2 8 5 5 6 6 7 4
     ------------------------------------------------------------------------
     9 3 9 7 7 0 2 3 8 3 5 7 0 3 4 8 3 4 4 0 3 0 7 0 7 0 0 6 5 8 6 6 9 3 1 9
     6 9 7 0 7 8 7 0 0 5 7 2 9 6 1 2 0 2 9 6 9 1 8 5 7 2 6 9 6 9 7 0 8 0 1 0
     5 4 5 6 7 2 8 8 6 8 7 2 7 8 1 6 7 2 9 5 4 9 0 0 6 8 9 8 0 7 7 7 8 0 5 8
     5 1 5 8 6 2 7 4 1 8 7 2 8 6 4 2 3 5 3 5 4 1 7 9 6 0 9 4 0 7 7 7 6 6 6 5
     4 5 0 4 8 3 4 7 0 4 8 8 1 7 4 9 9 4 7 1 3 2 9 9 6 1 8 8 3 6 5 9 4 1 3 0
     ------------------------------------------------------------------------
     2 7 0 1 0 3 6 4 5 9 5 3 3 1 1 6 7 4 7 7 9 1 6 9 2 7 8 6 0 7 5 2 3 3 6 1
     0 0 8 3 0 4 5 0 3 7 2 3 3 1 3 0 7 9 3 8 4 3 7 6 8 2 4 9 1 1 7 7 2 8 0 8
     5 2 2 3 6 7 0 6 8 3 3 3 3 6 3 9 2 5 0 0 1 9 4 5 1 3 8 0 5 2 6 5 2 0 0 9
     5 1 7 6 1 0 6 0 3 7 5 1 7 6 3 3 3 5 8 4 9 5 7 6 6 7 4 6 9 1 0 9 2 7 2 9
     3 7 4 6 8 4 2 1 8 4 5 7 3 5 4 0 7 8 1 4 8 8 0 4 9 8 3 6 9 4 5 1 1 8 1 3
     ------------------------------------------------------------------------
     2 0 3 8 3 4 5 1 1 1 2 2 3 8 1 9 8 1 0 8 5 1 5 4 2 8 1 2 9 0 0 5 8 6 5 2
     8 0 6 1 8 8 9 0 5 5 9 2 5 8 2 2 6 5 1 8 8 5 9 1 6 4 0 4 3 5 3 1 5 0 1 6
     1 5 5 0 8 1 0 2 3 0 0 8 4 7 3 2 0 8 2 0 4 8 5 8 9 1 7 0 0 7 3 3 3 9 3 3
     4 2 2 5 1 2 3 8 9 6 8 7 2 3 7 9 9 6 8 0 5 0 5 5 2 7 2 3 5 7 2 1 9 4 2 0
     4 0 3 1 8 3 3 4 7 0 3 4 1 6 3 5 1 1 1 8 0 0 3 7 2 1 7 0 5 6 9 3 3 3 5 1
     ------------------------------------------------------------------------
     4 3 7 1 0 4 1 |
     5 2 2 9 9 9 8 |
     1 6 1 4 1 3 8 |
     5 0 7 9 1 4 8 |
     0 1 4 3 5 6 3 |

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 6.0226 seconds

i8 : time C = points(M,R);
     -- used 0.764659 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` :