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

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = affinePointsByIntersection(M,R);
     -- used 3.86685 seconds

i8 : time C = affinePoints(M,R);
     -- used 0.686275 seconds

i9 : J == C_2  

o9 = true

Ways to use `affinePoints` :

affinePoints(Matrix,Ring)

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

Synopsis

Description

See also

Ways to use affinePoints :

Ways to use `affinePoints` :