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 = | 0 0 7 8 8 |
     | 9 1 4 9 9 |
     | 7 2 2 6 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                   
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - 2y - 9z +
                                                                           
     ------------------------------------------------------------------------
               40 2        35    304          2      2   15    45         
     18, x*z + --z  + 3x - --y - ---z + 161, y  + 10z  + --x - --y - 70z +
                3           3     3                       2     2         
     ------------------------------------------------------------------------
     243        70 2   17    175    490    525   2   14 2   9    35    98   
     ---, x*y + --z  + --x - ---y - ---z + ---, x  + --z  - -x - --y - --z +
      2          3      2     6      3      2         3     2     6     3   
     ------------------------------------------------------------------------
     105   3      2   15    35          483
     ---, z  - 19z  - --x + --y + 90z - ---})
      2                4     4           4

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 7.39102 seconds

i8 : time C = points(M,R);
     -- used 0.506012 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` :