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 = | 8 6 1 4 7 |
     | 7 2 6 6 1 |
     | 9 4 4 5 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          145 2   15 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  - --x
                                                                  256     16 
     ------------------------------------------------------------------------
       331    489    601         457 2   151    525    5023    3281   2  
     - ---y + ---z + ---, x*z - ----z  - ---x + ---y - ----z + ----, y  -
        64    256     64        1024      64    256    1024     256      
     ------------------------------------------------------------------------
     155 2    5    1049    1635    691         65  2   353    1381    2073   
     ---z  - --x - ----y + ----z + ---, x*y + ----z  - ---x - ----y - ----z +
     512     32     128     512    128        1024      64     256    1024   
     ------------------------------------------------------------------------
     9975   2   19 2        15    123    125   3   495 2   15    75    1783 
     ----, x  - --z  - 4x + --y + ---z - ---, z  - ---z  + --x + --y + ----z
      256       16           4     16     4         32      2     8     32  
     ------------------------------------------------------------------------
       825
     - ---})
        8

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 3.11453 seconds

i8 : time C = points(M,R);
     -- used 0.605908 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` :