macaulayFormula -- Macaulay formula for the resultant

Usage:

macaulayFormula F
Inputs:
- F, a matrix, a row matrix whose entries are $n+1$ homogeneous polynomials $F_0,\ldots,F_n$ in $n+1$ variables (or a list to be interpreted as such a matrix)
Outputs:
- a pair of two square matrices $(D,D')$ such that the ratio of their determinants $det(D)/det(D')$ is the resultant of $F_0,\ldots,F_n$

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 0.00493102s (cpu); 0.00478628s (thread); 0s (gc)

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   9       7 2   3       7        7 2   7 2   7       3 2          
o3 = {-p  + -p p  + -p  + -p p  + -p p  + --p , --p  + -p p  + -p  + 7p p  +
      2 0   4 0 1   4 1   4 0 2   9 1 2   10 2  10 0   3 0 1   7 1     0 2  
     ------------------------------------------------------------------------
     6         2  5 3   2 2     3   2      3    3 2               3 2    
     -p p  + 6p , -p  + -p p  + -p p  + 10p  + --p p  + 5p p p  + -p p  +
     7 1 2     2  4 0   9 0 1   7 0 1      1   10 0 2     0 1 2   2 1 2  
     ------------------------------------------------------------------------
     10   2   7   2   5 3
     --p p  + -p p  + -p }
      9 0 2   8 1 2   6 2

o3 : List

i4 : time (D,D') = macaulayFormula F
 -- used 0.00399615s (cpu); 0.00341832s (thread); 0s (gc)

o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0   
      | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0   
      | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0   
      | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0   
      | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0   
      | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
      | 0   0   0   0    0    0    9/2  0    0    0    9/4 3/4 0    0    0   
      | 0   0   0   0    0    0    0    9/2  0    0    0   9/4 3/4  0    0   
      | 0   0   0   0    0    0    0    0    9/2  0    0   0   9/4  3/4  0   
      | 0   0   0   0    0    0    0    0    0    9/2  0   0   0    9/4  3/4 
      | 0   0   0   7/10 0    0    7/3  7    0    0    3/7 6/7 6    0    0   
      | 0   0   0   0    7/10 0    0    7/3  7    0    0   3/7 6/7  6    0   
      | 0   0   0   0    0    7/10 0    0    7/3  7    0   0   3/7  6/7  6   
      | 0   5/4 0   2/9  3/10 0    3/7  5    10/9 0    10  3/2 7/8  5/6  0   
      | 0   0   5/4 0    2/9  3/10 0    3/7  5    10/9 0   10  3/2  7/8  5/6 
      | 0   0   0   0    0    0    7/10 0    0    0    7/3 7   0    0    0   
      | 0   0   0   0    0    0    0    7/10 0    0    0   7/3 7    0    0   
      | 0   0   0   0    0    0    0    0    7/10 0    0   0   7/3  7    0   
      | 0   0   0   0    0    0    0    0    0    7/10 0   0   0    7/3  7   
      | 0   0   0   0    5/4  0    0    2/9  3/10 0    0   3/7 5    10/9 0   
      | 0   0   0   0    0    5/4  0    0    2/9  3/10 0   0   3/7  5    10/9
     ------------------------------------------------------------------------
     0   0   0    0    0    0    |, | 9/2 9/4 3/4 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    7/10 |
     0   0   0    0    0    0    |  | 0   0   0   9/2  7/4  |
     0   0   0    0    0    0    |  | 0   0   0   7/10 3/7  |
     0   0   0    0    0    0    |
     7/4 7/9 7/10 0    0    0    |
     0   7/4 7/9  7/10 0    0    |
     0   0   7/4  7/9  7/10 0    |
     0   0   0    7/4  7/9  7/10 |
     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    |
     3/7 6/7 6    0    0    0    |
     0   3/7 6/7  6    0    0    |
     0   0   3/7  6/7  6    0    |
     0   0   0    3/7  6/7  6    |
     0   10  3/2  7/8  5/6  0    |
     0   0   10   3/2  7/8  5/6  |

o4 : Sequence

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

Ways to use macaulayFormula:

macaulayFormula(List)
macaulayFormula(Matrix)

For the programmer

The object macaulayFormula is a method function.

The source of this document is in Resultants.m2:982:0.

macaulayFormula -- Macaulay formula for the resultant

Description

See also

Ways to use macaulayFormula:

For the programmer