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 5 6 5 3 |
     | 7 3 3 0 5 |
     | 3 2 1 7 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          40 2   276 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  - ---x
                                                                  59      59 
     ------------------------------------------------------------------------
       473    573    3431         5 2   241    450    20    3165   2   26 2  
     - ---y - ---z + ----, x*z - --z  + ---x + ---y + --z - ----, y  - --z  -
        59     59     59         59      59     59    59     59        59    
     ------------------------------------------------------------------------
     210    787    132    3248         6 2   171    345    330    3459   2  
     ---x - ---y - ---z + ----, x*y + --z  + ---x + ---y + ---z - ----, x  -
      59     59     59     59         59      59     59     59     59       
     ------------------------------------------------------------------------
      9 2   1171    960    495    8286   3   594 2   240    360    1609   
     --z  - ----x - ---y - ---z + ----, z  - ---z  + ---x + ---y + ----z -
     59      59      59     59     59         59      59     59     59    
     ------------------------------------------------------------------------
     3594
     ----})
      59

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

              5        150
o6 : Matrix ZZ  <--- ZZ

i7 : time J = pointsByIntersection(M,R);
     -- used 6.87871 seconds

i8 : time C = points(M,R);
     -- used 1.00849 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` :