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 = | 6 5 3 5 2 |
     | 5 0 4 9 9 |
     | 5 6 2 9 2 |

              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          219 2  
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  -
                                                                   85    
     ------------------------------------------------------------------------
     168    338    2004    12         9  2   55    21    859    248   2  
     ---x - ---y + ----z + --, x*z + ---z  - --x - --y - ---z + ---, y  +
      17     85     85     17        170     17    85    170     17      
     ------------------------------------------------------------------------
     39 2   280    165    549    490        237 2   195    297    2787   
     --z  + ---x - ---y - ---z + ---, x*y - ---z  - ---x - ---y + ----z +
     17      17     17     17     17        170      17     85     170   
     ------------------------------------------------------------------------
     156   2    57 2   133    48    567    375   3   1427 2   84    84   
     ---, x  + ---z  - ---x - --y - ---z + ---, z  - ----z  - --x - --y +
      17       170      17    85    170     17        85      17    85   
     ------------------------------------------------------------------------
     7122    1524
     ----z - ----})
      85      17

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 8.08739 seconds

i8 : time C = points(M,R);
     -- used 1.18604 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` :