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Resultants :: macaulayFormula

macaulayFormula -- Macaulay formula for the resultant

Synopsis

Description

This formula is stated in Theorem 4.9 of Using Algebraic Geometry, by David A. Cox, John Little, Donal O'shea.

i1 : F = genericPolynomials {2,2,3}

         2               2                        2     2               2  
o1 = {a x  + a x x  + a x  + a x x  + a x x  + a x , b x  + b x x  + b x  +
       0 0    1 0 1    3 1    2 0 2    4 1 2    5 2   0 0    1 0 1    3 1  
     ------------------------------------------------------------------------
                          2     3      2          2      3      2    
     b x x  + b x x  + b x , c x  + c x x  + c x x  + c x  + c x x  +
      2 0 2    4 1 2    5 2   0 0    1 0 1    3 0 1    6 1    2 0 2  
     ------------------------------------------------------------------------
                   2          2        2      3
     c x x x  + c x x  + c x x  + c x x  + c x }
      4 0 1 2    7 1 2    5 0 2    8 1 2    9 2

o1 : List
i2 : time (D,D') = macaulayFormula F
     -- used 0.00269986 seconds

o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
      | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
      | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
      | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
      | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
      | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
      | 0   0   0   0   0   0   a_0 0   0   0   a_1 a_2 0   0   0   a_3 a_4
      | 0   0   0   0   0   0   0   a_0 0   0   0   a_1 a_2 0   0   0   a_3
      | 0   0   0   0   0   0   0   0   a_0 0   0   0   a_1 a_2 0   0   0  
      | 0   0   0   0   0   0   0   0   0   a_0 0   0   0   a_1 a_2 0   0  
      | 0   0   0   b_0 0   0   b_1 b_2 0   0   b_3 b_4 b_5 0   0   0   0  
      | 0   0   0   0   b_0 0   0   b_1 b_2 0   0   b_3 b_4 b_5 0   0   0  
      | 0   0   0   0   0   b_0 0   0   b_1 b_2 0   0   b_3 b_4 b_5 0   0  
      | 0   c_0 0   c_1 c_2 0   c_3 c_4 c_5 0   c_6 c_7 c_8 c_9 0   0   0  
      | 0   0   c_0 0   c_1 c_2 0   c_3 c_4 c_5 0   c_6 c_7 c_8 c_9 0   0  
      | 0   0   0   0   0   0   b_0 0   0   0   b_1 b_2 0   0   0   b_3 b_4
      | 0   0   0   0   0   0   0   b_0 0   0   0   b_1 b_2 0   0   0   b_3
      | 0   0   0   0   0   0   0   0   b_0 0   0   0   b_1 b_2 0   0   0  
      | 0   0   0   0   0   0   0   0   0   b_0 0   0   0   b_1 b_2 0   0  
      | 0   0   0   0   c_0 0   0   c_1 c_2 0   0   c_3 c_4 c_5 0   0   c_6
      | 0   0   0   0   0   c_0 0   0   c_1 c_2 0   0   c_3 c_4 c_5 0   0  
     ------------------------------------------------------------------------
     0   0   0   0   |, | a_0 a_1 a_2 0   0   |)
     0   0   0   0   |  | 0   a_0 0   0   0   |
     0   0   0   0   |  | 0   0   a_0 0   a_5 |
     0   0   0   0   |  | 0   0   0   a_0 a_3 |
     0   0   0   0   |  | 0   0   0   b_0 b_3 |
     0   0   0   0   |
     a_5 0   0   0   |
     a_4 a_5 0   0   |
     a_3 a_4 a_5 0   |
     0   a_3 a_4 a_5 |
     0   0   0   0   |
     0   0   0   0   |
     0   0   0   0   |
     0   0   0   0   |
     0   0   0   0   |
     b_5 0   0   0   |
     b_4 b_5 0   0   |
     b_3 b_4 b_5 0   |
     0   b_3 b_4 b_5 |
     c_7 c_8 c_9 0   |
     c_6 c_7 c_8 c_9 |

o2 : Sequence
i3 : F = {random(2,Grass(0,2)),random(2,Grass(0,2)),random(3,Grass(0,2))}

      9 2   1       1 2   9              3 2  3 2   3       7 2   7      
o3 = {-p  + -p p  + -p  + -p p  + p p  + -p , -p  + -p p  + -p  + -p p  +
      2 0   2 0 1   2 1   4 0 2    1 2   4 2  2 0   4 0 1   9 1   4 0 2  
     ------------------------------------------------------------------------
      7       1 2   7 3   7 2     3   2   2 3     2     5          2    
     --p p  + -p , --p  + -p p  + -p p  + -p  + 7p p  + -p p p  + p p  +
     10 1 2   2 2  10 0   3 0 1   7 0 1   3 1     0 2   2 0 1 2    1 2  
     ------------------------------------------------------------------------
     6   2       2     3
     -p p  + 2p p  + 6p }
     7 0 2     1 2     2

o3 : List
i4 : time (D,D') = macaulayFormula F
     -- used 0.0021609 seconds

o4 = (| 9/2 1/2  9/4  1/2 1    3/4  0   0   0   0   0   0    0    0    0  
      | 0   9/2  0    1/2 9/4  0    1/2 1   3/4 0   0   0    0    0    0  
      | 0   0    9/2  0   1/2  9/4  0   1/2 1   3/4 0   0    0    0    0  
      | 0   0    0    9/2 0    0    1/2 9/4 0   0   1/2 1    3/4  0    0  
      | 0   0    0    0   9/2  0    0   1/2 9/4 0   0   1/2  1    3/4  0  
      | 0   0    0    0   0    9/2  0   0   1/2 9/4 0   0    1/2  1    3/4
      | 0   0    0    0   0    0    9/2 0   0   0   1/2 9/4  0    0    0  
      | 0   0    0    0   0    0    0   9/2 0   0   0   1/2  9/4  0    0  
      | 0   0    0    0   0    0    0   0   9/2 0   0   0    1/2  9/4  0  
      | 0   0    0    0   0    0    0   0   0   9/2 0   0    0    1/2  9/4
      | 0   0    0    3/2 0    0    3/4 7/4 0   0   7/9 7/10 1/2  0    0  
      | 0   0    0    0   3/2  0    0   3/4 7/4 0   0   7/9  7/10 1/2  0  
      | 0   0    0    0   0    3/2  0   0   3/4 7/4 0   0    7/9  7/10 1/2
      | 0   7/10 0    7/3 7    0    3/7 5/2 6/7 0   2/3 1    2    6    0  
      | 0   0    7/10 0   7/3  7    0   3/7 5/2 6/7 0   2/3  1    2    6  
      | 0   0    0    0   0    0    3/2 0   0   0   3/4 7/4  0    0    0  
      | 0   0    0    0   0    0    0   3/2 0   0   0   3/4  7/4  0    0  
      | 0   0    0    0   0    0    0   0   3/2 0   0   0    3/4  7/4  0  
      | 0   0    0    0   0    0    0   0   0   3/2 0   0    0    3/4  7/4
      | 0   0    0    0   7/10 0    0   7/3 7   0   0   3/7  5/2  6/7  0  
      | 0   0    0    0   0    7/10 0   0   7/3 7   0   0    3/7  5/2  6/7
     ------------------------------------------------------------------------
     0   0    0    0    0    0   |, | 9/2 1/2 9/4 0   0   |)
     0   0    0    0    0    0   |  | 0   9/2 0   0   0   |
     0   0    0    0    0    0   |  | 0   0   9/2 0   3/4 |
     0   0    0    0    0    0   |  | 0   0   0   9/2 1/2 |
     0   0    0    0    0    0   |  | 0   0   0   3/2 7/9 |
     0   0    0    0    0    0   |
     1/2 1    3/4  0    0    0   |
     0   1/2  1    3/4  0    0   |
     0   0    1/2  1    3/4  0   |
     0   0    0    1/2  1    3/4 |
     0   0    0    0    0    0   |
     0   0    0    0    0    0   |
     0   0    0    0    0    0   |
     0   0    0    0    0    0   |
     0   0    0    0    0    0   |
     7/9 7/10 1/2  0    0    0   |
     0   7/9  7/10 1/2  0    0   |
     0   0    7/9  7/10 1/2  0   |
     0   0    0    7/9  7/10 1/2 |
     0   2/3  1    2    6    0   |
     0   0    2/3  1    2    6   |

o4 : Sequence
i5 : assert(det D == (resultant F) * (det D'))

See also

Ways to use macaulayFormula :

For the programmer

The object macaulayFormula is a method function.