next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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 = | 4 7 1 2 6 |
     | 8 8 7 8 2 |
     | 1 2 5 4 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          33 2   18 
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z + --z  + --x
                                                                  43     43 
     ------------------------------------------------------------------------
       89    497    760        119 2   80     1    309    1276   2   55 2  
     - --y - ---z + ---, x*z + ---z  - --x - --y - ---z + ----, y  + --z  +
       43     43     43        129     43    43     43     129       43    
     ------------------------------------------------------------------------
     30    435    255    808        55 2   374    253    255    1944   2  
     --x - ---y - ---z + ---, x*y - --z  - ---x - ---y + ---z + ----, x  -
     43     43     43     43        43      43     43     43     43       
     ------------------------------------------------------------------------
     160 2   463    16    130    3646   3   433 2   72    12    1214    632
     ---z  - ---x - --y + ---z + ----, z  - ---z  - --x + --y + ----z - ---})
     129      43    43     43     129        43     43    43     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 4 4 8 1 5 4 7 9 7 4 8 7 8 1 0 5 4 3 7 1 9 5 2 9 9 8 8 2 6 3 8 4 5 0
     | 4 5 1 9 1 0 1 8 7 3 6 3 2 1 3 0 4 6 8 1 5 6 0 6 2 9 4 5 9 3 2 7 7 5 8
     | 7 3 3 5 3 4 4 7 0 6 9 5 1 9 8 2 5 9 8 9 3 9 1 5 5 5 3 7 0 3 9 5 6 4 4
     | 3 9 9 4 4 6 0 8 5 4 8 1 2 6 9 6 9 9 6 8 4 6 5 5 7 4 3 5 8 3 9 1 3 2 6
     | 1 9 0 7 9 4 5 1 6 5 3 3 6 9 7 5 9 3 5 1 0 1 7 1 7 3 7 0 8 6 4 5 1 0 1
     ------------------------------------------------------------------------
     3 6 4 2 4 2 3 3 7 6 7 2 5 0 4 2 2 7 5 4 7 9 2 6 1 9 3 7 7 4 0 9 8 5 8 7
     8 3 6 0 8 6 5 2 0 5 7 3 3 9 9 0 9 1 5 0 4 2 8 5 3 9 9 6 7 8 2 6 3 9 4 9
     2 1 1 5 1 7 4 0 7 1 1 1 6 3 1 9 3 3 9 9 6 4 1 4 0 1 0 8 3 6 5 3 6 9 2 8
     1 7 6 0 1 6 4 6 0 0 3 9 5 7 2 0 9 8 1 7 5 4 6 7 2 1 2 3 1 1 5 5 7 0 1 9
     6 5 8 2 3 7 1 4 5 0 3 1 3 7 2 3 3 1 9 7 9 0 2 8 9 0 2 3 1 0 9 6 0 6 4 9
     ------------------------------------------------------------------------
     4 0 6 2 7 1 6 2 6 3 9 6 9 1 3 1 2 0 2 1 6 2 5 4 6 4 1 7 5 8 3 1 8 1 2 4
     5 2 7 7 7 8 6 9 1 1 1 8 1 5 8 8 0 0 5 2 0 8 6 5 4 0 5 8 1 5 4 8 3 0 0 9
     3 2 8 6 3 5 4 4 4 2 6 9 6 7 6 2 2 1 8 6 8 1 4 3 9 3 5 1 0 2 5 9 3 6 2 4
     0 6 9 7 5 8 1 1 0 1 1 7 5 8 2 4 1 0 2 0 3 3 9 6 0 7 2 5 7 7 4 1 8 0 0 1
     5 0 3 5 6 1 7 1 3 1 0 1 2 9 0 0 2 9 2 5 9 8 4 4 8 1 9 7 2 4 0 6 1 2 5 4
     ------------------------------------------------------------------------
     5 0 6 7 3 9 5 6 0 0 2 1 0 2 0 5 5 7 1 9 1 9 7 9 8 9 2 9 1 6 6 9 8 8 8 0
     2 9 8 7 8 1 5 8 3 5 1 7 7 6 0 7 9 2 5 5 0 4 7 1 8 3 6 2 1 2 4 6 6 5 0 3
     0 8 5 7 6 3 8 8 4 9 5 3 8 1 3 8 8 6 8 7 6 5 2 4 2 1 5 6 3 4 8 4 9 1 1 2
     0 3 9 9 0 7 5 6 4 6 5 3 4 5 8 9 1 5 8 7 0 6 3 9 4 4 8 2 0 3 0 3 0 4 1 9
     0 5 8 7 3 4 6 5 8 0 7 8 0 7 1 4 4 5 2 7 0 8 7 7 2 2 6 1 3 3 5 9 2 4 5 2
     ------------------------------------------------------------------------
     5 2 8 8 4 9 8 |
     0 5 6 9 3 5 2 |
     3 8 4 8 5 6 4 |
     3 5 9 9 9 2 9 |
     9 9 8 6 9 7 4 |

              5        150
o6 : Matrix ZZ  <--- ZZ
i7 : time J = pointsByIntersection(M,R);
     -- used 26.6505 seconds
i8 : time C = points(M,R);
     -- used 1.79327 seconds
i9 : J == C_2  

o9 = true

See also

Ways to use points :