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Points :: points

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

Synopsis

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 9 7 5 7 |
     | 7 1 0 6 9 |
     | 2 0 1 0 3 |

              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          707 2   75 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - ---z  + --x
                                                                   86     43 
     ------------------------------------------------------------------------
       60    1127    735        448 2   210    168    1337    2058   2  
     + --y + ----z - ---, x*z + ---z  - ---x - ---y - ----z + ----, y  +
       43     86      43         43      43     43     43      43       
     ------------------------------------------------------------------------
     213 2   120    397    807    1434        909 2   543    615    2223   
     ---z  - ---x - ---y - ---z + ----, x*y + ---z  - ---x - ---y - ----z +
      43      43     43     43     43          43      43     43     43    
     ------------------------------------------------------------------------
     5115   2   1668 2   153    604    3954    5464   3   108 2   30    24   
     ----, x  - ----z  + ---x + ---y + ----z - ----, z  - ---z  - --x - --y -
      43         43       43     43     43      43         43     43    43   
     ------------------------------------------------------------------------
     19    294
     --z + ---})
     43     43

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

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 4.71662 seconds
i8 : time C = points(M,R);
     -- used 0.426714 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :