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 0 1 0 2 |
     | 4 9 4 2 1 |
     | 2 6 7 6 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            2       
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + 3z  - 6x -
     ------------------------------------------------------------------------
                             2                     2     2                   
     6y - 37z + 114, x*z + 3z  - 10x - 36z + 108, y  + 4z  - 3x - 11y - 39z +
     ------------------------------------------------------------------------
                  2                   2     2                    3     2
     108, x*y - 3z  + 2x + 33z - 90, x  - 5z  + 6x + 58z - 168, z  - 9z  -
     ------------------------------------------------------------------------
     12x + 2z + 96})

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 11.7312 seconds

i8 : time C = points(M,R);
     -- used 0.99161 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` :