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 = | 6 9 4 9 1 |
     | 0 7 1 1 3 |
     | 0 6 5 1 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          109 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + ---z  +
                                                                  217    
     ------------------------------------------------------------------------
     604    2022    116    3624        1081 2   1895    4005    4714   
     ---x - ----y - ---z - ----, x*z + ----z  + ----x - ----y - ----z -
     217     217    217     217         217      217     217     217   
     ------------------------------------------------------------------------
     11370   2   75 2   204    523    195    1224        709 2   1275   
     -----, y  + --z  + ---x - ---y - ---z - ----, x*y + ---z  + ----x -
      217        31      31     31     31     31         217      217   
     ------------------------------------------------------------------------
     4098    2389    7650   2   1110 2   5245    3450    3630    23658   3  
     ----y - ----z - ----, x  - ----z  - ----x + ----y + ----z + -----, z  -
      217     217     217        217      217     217     217     217       
     ------------------------------------------------------------------------
     3996 2   4560    5910    11549    27360
     ----z  - ----x + ----y + -----z + -----})
      217      217     217     217      217

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 3.18052 seconds

i8 : time C = points(M,R);
     -- used 0.842872 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` :