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 3 1 9 4 |
     | 4 4 1 7 7 |
     | 5 1 2 4 1 |

              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          1 2       
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - -z  - 2x -
                                                                  3         
     ------------------------------------------------------------------------
     1         17        4 2   13    10    17    97   2     2             
     -y - 2z + --, x*z + -z  - --x + --y - --z + --, y  + 5z  + 3x - 12y -
     3          3        9      3     9     3     9                       
     ------------------------------------------------------------------------
                     10 2        11          103   2   10 2         4       
     30z + 48, x*y + --z  - 5x - --y - 20z + ---, x  + --z  - 11x + -y - 20z
                      3           3           3         3           3       
     ------------------------------------------------------------------------
       106   3   25 2       1          40
     + ---, z  - --z  + x - -y + 19z - --})
        3         3         3           3

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 11.3883 seconds

i8 : time C = points(M,R);
     -- used 1.0106 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` :