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 = | 5 8 9 6 9 |
     | 0 9 4 0 6 |
     | 9 2 8 6 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          11 2   69 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  - --x
                                                                  26     13 
     ------------------------------------------------------------------------
       38    211    849         69 2   1128     4     1840    10176   2  
     - --y - ---z + ---, x*z + ---z  - ----x - ---y - ----z + -----, y  +
       13     26     13        143      143    143     143     143       
     ------------------------------------------------------------------------
     223 2   2175    1541    1895    11379         81 2   513    1062    873 
     ---z  + ----x - ----y - ----z - -----, x*y - ---z  - ---x - ----y + ---z
     572      286     143     572     286         572     286     143    572 
     ------------------------------------------------------------------------
       1917   2    49 2   2074     18    401    6975   3   2029 2   2232   
     + ----, x  - ---z  - ----x + ---y + ---z + ----, z  - ----z  + ----x -
        286       286      143    143    286     143        143      143   
     ------------------------------------------------------------------------
     1416    6726    11592
     ----y + ----z - -----})
      143     143     143

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 10.9162 seconds

i8 : time C = points(M,R);
     -- used 1.00707 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` :