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 = | 0 2 1 0 8 |
     | 0 1 7 2 2 |
     | 0 6 6 6 5 |

              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          2 2       
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - -z  - 6y +
                                                                  5         
     ------------------------------------------------------------------------
     12         8 2        48    2   791 2   30    93    5101         36 2  
     --z, x*z - -z  - 6x + --z, y  - ---z  - --x - --y + ----z, x*y + --z  -
      5         5           5        165     11    11     165         55    
     ------------------------------------------------------------------------
     17    12    196    2   322 2   21     2    1934    3      2
     --x - --y - ---z, x  + ---z  - --x + --y - ----z, z  - 11z  + 30z})
     11    11     55         33     11    11     33

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 22.2905 seconds

i8 : time C = points(M,R);
     -- used 1.92744 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` :