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 5 2 5 1 |
     | 5 4 7 8 8 |
     | 4 7 3 3 4 |

              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          5 2    3   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + -z  + --x -
                                                                  6     14   
     ------------------------------------------------------------------------
     51    545    71        1 2   47    15    15    11   2   3    96    12   
     --y - ---z + --, x*z - -z  - --x + --y + --z - --, y  - -x - --y - --z +
     14     42     2        2     14    14    14     2       7     7     7   
     ------------------------------------------------------------------------
                2   50    32    73         2   4 2        40         3      2
     53, x*y - z  - --x - --y + --z + 10, x  + -z  - 7x - --z + 38, z  - 14z 
                     7     7     7             3           3
     ------------------------------------------------------------------------
     + 61z - 84})

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 11.0776 seconds

i8 : time C = points(M,R);
     -- used 1.01227 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` :