affinePoints -- produces the ideal and initial ideal from the coordinates of a finite set of points

Synopsis

Usage:

(Q,inG,G) = affinePoints(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 7 3 8 8 |
     | 1 8 7 5 5 |
     | 3 3 8 7 2 |

              3       5
o1 : Matrix ZZ  <-- ZZ

i2 : R = QQ[x,y,z]

o2 = R

o2 : PolynomialRing

i3 : (Q,inG,G) = affinePoints(M,R)

                    2                     2        2   3          10 2   70 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - --z  + --x
                                                                  39     39 
     ------------------------------------------------------------------------
       107    35    55        25 2   292    25    179    937   2   31 2   56 
     - ---y - --z - --, x*z + --z  - ---x - --y - ---z + ---, y  + --z  + --x
        39    13    13        39      39    39     13     13       13     13 
     ------------------------------------------------------------------------
       109    279    206         5 2   277    307    15    707   2   20 2  
     - ---y - ---z + ---, x*y - --z  - ---x - ---y + --z + ---, x  - --z  -
        13     13     13        39      39     39    13     13       39    
     ------------------------------------------------------------------------
     445    20    60    228   3   166 2   70    10    623    1296
     ---x + --y + --z + ---, z  - ---z  + --x + --y + ---z - ----})
      39    39    13     13        13     13    13     13     13

o3 : Sequence

i4 : monomialIdeal G == inG

o4 = true

The Buchberger-Moeller algorithm in points may be faster than the alternative method using the intersection of the ideals for each point.

Ways to use affinePoints :

affinePoints(Matrix,Ring)

For the programmer

The object affinePoints is a method function.

affinePoints -- produces the ideal and initial ideal from the coordinates of a finite set of points

Synopsis

Description

See also

Ways to use affinePoints :

For the programmer