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 = | 7 7 4 0 1 |
     | 1 4 3 5 4 |
     | 7 1 0 4 7 |

              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           8 2   21 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + --x
                                                                  17     34 
     ------------------------------------------------------------------------
       98    13    252        21 2   35    168    203    364   2    7 2    5 
     - --y + --z + ---, x*z + --z  - --x + ---y - ---z - ---, y  + --z  + --x
       17    34     17        17     17     17     17     17       17     34 
     ------------------------------------------------------------------------
       80    107    77        12 2   147    130    181    480   2   15 2  
     - --y - ---z + --, x*y - --z  - ---x - ---y + ---z + ---, x  + --z  -
       17     34    17        17      34     17     34     17       17    
     ------------------------------------------------------------------------
     144    16    128    352   3   190 2   63    126    488    630
     ---x - --y - ---z + ---, z  - ---z  - --x - ---y + ---z + ---})
      17    17     17     17        17     17     17     17     17

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 4.6114 seconds

i8 : time C = points(M,R);
     -- used 0.515689 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` :