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 = | 3 5 5 3 1 |
     | 0 5 2 7 4 |
     | 2 7 7 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          77 2   525 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  + ---x
                                                                  16      16 
     ------------------------------------------------------------------------
            903    77         7 2    7    27     7   2   11 2   85        
     - 7y - ---z - --, x*z - --z  - --x - --z + --, y  + --z  - --x - 7y -
             16    16        16     16    16    16       16     16        
     ------------------------------------------------------------------------
     33    277        29 2   125         311    131   2   1 2   17    3   
     --z + ---, x*y + --z  + ---x - 5y - ---z + ---, x  - -z  - --x + -z +
     16     16        16      16          16     16       4      4    4   
     ------------------------------------------------------------------------
     13   3   153 2   105    347    105
     --, z  - ---z  - ---x + ---z + ---})
      4        16      16     16     16

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 6.7983 seconds

i8 : time C = points(M,R);
     -- used 0.659174 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` :