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Resultants :: resultant

resultant -- multipolynomial resultant

Synopsis

Description

Let F0,...,Fn be n+1 homogeneous polynomials in n+1 variables x0,...,xn over a commutative ring K. The resultant R(F0,...,Fn) is a certain polynomial in the coefficients of F0,...,Fn; when K is an algebraically closed field, R(F0,...,Fn) vanishes if and only if F0,...,Fn have a common nontrivial root. For the general theory, see one of the following: Using Algebraic Geometry, by David A. Cox, John Little, Donal O'shea; Discriminants, Resultants, and Multidimensional Determinants, by Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky.

i1 : ZZ[t,u][x,y,z]

o1 = ZZ[t, u][x, y, z]

o1 : PolynomialRing
i2 : F = {x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4}

       2               2                            3      2         4
o2 = {x  + 3t*y*z - u*z , (t + 3u - 1)x - y, - t*x*y  + t*x y*z + u*z }

o2 : List
i3 : time resultant F
     -- used 0.0145034 seconds

          12         11 2         10 3         9 4          8 5          7 6
o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
     ------------------------------------------------------------------------
              6 7          5 8       11          10 2         9 3  
     - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
     ------------------------------------------------------------------------
            8 4          7 5          6 6          5 7        4 8        3 9
     306181t u  + 688923t u  + 826821t u  + 413883t u  + 1215t u  + 1458t u 
     ------------------------------------------------------------------------
           2 10        10          9 2          8 3          7 4          6 5
     + 729t u   - 1701t  u - 25515t u  - 153093t u  - 459321t u  - 689265t u 
     ------------------------------------------------------------------------
              5 6        4 7        3 8        2 9        9          8 2  
     - 414693t u  - 2835t u  - 3159t u  - 1458t u  + 2835t u + 33984t u  +
     ------------------------------------------------------------------------
            7 3          6 4          5 5         4 6        3 7        2 8  
     152565t u  + 303135t u  + 220905t u  - 12150t u  - 5913t u  + 1215t u  -
     ------------------------------------------------------------------------
          8          7 2         6 3         5 4         4 5         3 6  
     2862t u - 25659t u  - 75873t u  - 70031t u  + 16263t u  + 13230t u  -
     ------------------------------------------------------------------------
         2 7        7          6 2         5 3        4 4        3 5  
     540t u  + 1809t u + 10818t u  + 15015t u  - 6609t u  - 9360t u  +
     ------------------------------------------------------------------------
         2 6       6         5 2       4 3        3 4      2 5        6  
     135t u  - 729t u - 2313t u  + 684t u  + 3165t u  - 81t u  - 54t*u  +
     ------------------------------------------------------------------------
         5        4 2       3 3      2 4        5      4       3 2      2 3  
     189t u + 135t u  - 583t u  - 38t u  + 54t*u  - 27t u + 54t u  + 47t u  -
     ------------------------------------------------------------------------
          4     2 2       3    2
     18t*u  - 9t u  + 2t*u  + u

o3 : ZZ[t, u]

Below we compute the general expression of the resultant of a linear, a quadratic and a cubic form on 2.

i4 : F = genericPolynomials({1,2,3},ZZ)

                             2               2                        2     3
o4 = {a x  + a x  + a x , b x  + b x x  + b x  + b x x  + b x x  + b x , c x 
       0 0    1 1    2 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2   0 0
     ------------------------------------------------------------------------
          2          2      3      2                   2          2        2
     + c x x  + c x x  + c x  + c x x  + c x x x  + c x x  + c x x  + c x x 
        1 0 1    3 0 1    6 1    2 0 2    4 0 1 2    7 1 2    5 0 2    8 1 2
     ------------------------------------------------------------------------
          3
     + c x }
        9 2

o4 : List
i5 : time resultant F
     -- used 3.53813 seconds

      6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
      2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
     ------------------------------------------------------------------------
       3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
     6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
       1 2 3 4 5 0     1 2 4 5 0     1 2 3 5 0     1 2 4 5 0    1 5 0  
     ------------------------------------------------------------------------
      6   2          5   2           5               2 4            
     a b b c c  + a a b b c c  + 2a a b b b c c  - 2a a b b b c c  +
      2 1 3 0 1    1 2 2 3 0 1     1 2 1 3 4 0 1     1 2 2 3 4 0 1  
     ------------------------------------------------------------------------
        5 2          2 4   2        3 3   2             4   2      
     a a b b c c  - a a b b c c  + a a b b c c  - 2a a a b b c c  +
      0 2 3 4 0 1    1 2 1 4 0 1    1 2 2 4 0 1     0 1 2 3 4 0 1  
     ------------------------------------------------------------------------
        2 3 3         2 4               3 3                   4 2        
     a a a b c c  - 2a a b b b c c  + 2a a b b b c c  - 2a a a b b c c  +
      0 1 2 4 0 1     1 2 1 3 5 0 1     1 2 2 3 5 0 1     0 1 2 3 5 0 1  
     ------------------------------------------------------------------------
       3 3               4 2                 2 3                 3 2 2      
     2a a b b b c c  - 2a a b b b c c  + 6a a a b b b c c  - 4a a a b b c c 
       1 2 1 4 5 0 1     1 2 2 4 5 0 1     0 1 2 3 4 5 0 1     0 1 2 4 5 0 1
     ------------------------------------------------------------------------
        4 2   2        5     2           3 2   2           4     2      
     - a a b b c c  + a a b b c c  - 4a a a b b c c  + 5a a a b b c c  -
        1 2 1 5 0 1    1 2 2 5 0 1     0 1 2 3 5 0 1     0 1 2 4 5 0 1  
     ------------------------------------------------------------------------
         5 3        6   2 2      5   2 2       5       2         4       2  
     2a a b c c  + a b b c  - a a b b c  - 2a a b b b c  + 2a a a b b b c  +
       0 1 5 0 1    2 0 3 1    0 2 2 3 1     1 2 0 3 4 1     0 1 2 2 3 4 1  
     ------------------------------------------------------------------------
      2 4   2 2      2 3   2 2     2 4       2       2 3       2    2 4 2   2
     a a b b c  - a a a b b c  + 2a a b b b c  - 2a a a b b b c  + a a b b c 
      1 2 0 4 1    0 1 2 2 4 1     1 2 0 3 5 1     0 1 2 2 3 5 1    0 2 3 5 1
     ------------------------------------------------------------------------
         3 3       2       3 2       2     2   3       2    2 2 2 2   2  
     - 2a a b b b c  + 2a a a b b b c  - 2a a a b b b c  + a a a b b c  +
         1 2 0 4 5 1     0 1 2 2 4 5 1     0 1 2 3 4 5 1    0 1 2 4 5 1  
     ------------------------------------------------------------------------
      4 2   2 2      4     2 2     2 2 2   2 2     2 3     2 2    2 4 3 2  
     a a b b c  - a a a b b c  + 2a a a b b c  - 2a a a b b c  + a a b c  +
      1 2 0 5 1    0 1 2 2 5 1     0 1 2 3 5 1     0 1 2 4 5 1    0 1 5 1  
     ------------------------------------------------------------------------
        5   2        2 4   2           5 3         2 4            
     a a b b c c  - a a b b c c  - 2a a b c c  - 2a a b b b c c  +
      1 2 1 3 0 2    1 2 2 3 0 2     0 2 3 0 2     1 2 1 3 4 0 2  
     ------------------------------------------------------------------------
       3 3                   4 2          3 3   2        4 2   2      
     2a a b b b c c  + 5a a a b b c c  + a a b b c c  - a a b b c c  -
       1 2 2 3 4 0 2     0 1 2 3 4 0 2    1 2 1 4 0 2    1 2 2 4 0 2  
     ------------------------------------------------------------------------
         2 3   2          3 2 3         3 3               4 2            
     4a a a b b c c  + a a a b c c  + 2a a b b b c c  - 2a a b b b c c  -
       0 1 2 3 4 0 2    0 1 2 4 0 2     1 2 1 3 5 0 2     1 2 2 3 5 0 2  
     ------------------------------------------------------------------------
         2 3 2           4 2               5                   3 2          
     4a a a b b c c  - 2a a b b b c c  + 2a a b b b c c  + 6a a a b b b c c 
       0 1 2 3 5 0 2     1 2 1 4 5 0 2     1 2 2 4 5 0 2     0 1 2 3 4 5 0 2
     ------------------------------------------------------------------------
           4   2          5     2        6   2           4     2      
     - 2a a a b b c c  + a a b b c c  - a b b c c  - 2a a a b b c c  +
         0 1 2 4 5 0 2    1 2 1 5 0 2    1 2 5 0 2     0 1 2 3 5 0 2  
     ------------------------------------------------------------------------
        5   2           5   2          5   2            4   2      
     a a b b c c  - 2a a b b c c  + a a b b c c  + a a a b b c c  +
      0 1 4 5 0 2     1 2 0 3 1 2    0 2 1 3 1 2    0 1 2 2 3 1 2  
     ------------------------------------------------------------------------
       2 4                   4                 2 3              2 4 2        
     4a a b b b c c  - 2a a a b b b c c  - 2a a a b b b c c  - a a b b c c  -
       1 2 0 3 4 1 2     0 1 2 1 3 4 1 2     0 1 2 2 3 4 1 2    0 2 3 4 1 2  
     ------------------------------------------------------------------------
       3 3   2          2 3   2          3 2   2         2   3   2      
     2a a b b c c  + a a a b b c c  + a a a b b c c  + 2a a a b b c c  -
       1 2 0 4 1 2    0 1 2 1 4 1 2    0 1 2 2 4 1 2     0 1 2 3 4 1 2  
     ------------------------------------------------------------------------
      2 2 2 3         3 3                 2 3                 3 2            
     a a a b c c  - 4a a b b b c c  + 2a a a b b b c c  + 2a a a b b b c c  +
      0 1 2 4 1 2     1 2 0 3 5 1 2     0 1 2 1 3 5 1 2     0 1 2 2 3 5 1 2  
     ------------------------------------------------------------------------
       4 2                 3 2                 4              
     4a a b b b c c  - 2a a a b b b c c  - 2a a a b b b c c  -
       1 2 0 4 5 1 2     0 1 2 1 4 5 1 2     0 1 2 2 4 5 1 2  
     ------------------------------------------------------------------------
       2 2 2               2 3   2           5     2          4     2      
     2a a a b b b c c  + 2a a a b b c c  - 2a a b b c c  + a a a b b c c  +
       0 1 2 3 4 5 1 2     0 1 2 4 5 1 2     1 2 0 5 1 2    0 1 2 1 5 1 2  
     ------------------------------------------------------------------------
        5   2        2 4   2        2 4   2 2        4   2 2    2 4 3 2  
     a a b b c c  - a a b b c c  + a a b b c  - a a a b b c  + a a b c  -
      0 1 2 5 1 2    0 1 4 5 1 2    1 2 0 3 2    0 1 2 1 3 2    0 2 3 2  
     ------------------------------------------------------------------------
       3 3       2       2 3       2     2   3 2   2    4 2   2 2  
     2a a b b b c  + 2a a a b b b c  - 2a a a b b c  + a a b b c  -
       1 2 0 3 4 2     0 1 2 1 3 4 2     0 1 2 3 4 2    1 2 0 4 2  
     ------------------------------------------------------------------------
        3 2   2 2    2 2 2   2 2     4 2       2       3 2       2  
     a a a b b c  + a a a b b c  + 2a a b b b c  - 2a a a b b b c  +
      0 1 2 1 4 2    0 1 2 3 4 2     1 2 0 3 5 2     0 1 2 1 3 5 2  
     ------------------------------------------------------------------------
       2 2 2 2   2     5         2       4         2     2 3         2  
     2a a a b b c  - 2a a b b b c  + 2a a a b b b c  - 2a a a b b b c  +
       0 1 2 3 5 2     1 2 0 4 5 2     0 1 2 1 4 5 2     0 1 2 3 4 5 2  
     ------------------------------------------------------------------------
      6   2 2      5   2 2    2 4   2 2    6 2             5            
     a b b c  - a a b b c  + a a b b c  + a b b c c  - 2a a b b b c c  +
      1 0 5 2    0 1 1 5 2    0 1 3 5 2    2 1 3 0 3     1 2 1 2 3 0 3  
     ------------------------------------------------------------------------
      2 4 2           6   2           5   2          5 2        
     a a b b c c  - 2a b b c c  + 2a a b b c c  - a a b b c c  +
      1 2 2 3 0 3     2 0 3 0 3     0 2 2 3 0 3    1 2 1 4 0 3  
     ------------------------------------------------------------------------
       2 4              3 3 2             5                 5            
     2a a b b b c c  - a a b b c c  + 4a a b b b c c  - 2a a b b b c c  -
       1 2 1 2 4 0 3    1 2 2 4 0 3     1 2 0 3 4 0 3     0 2 1 3 4 0 3  
     ------------------------------------------------------------------------
           4               2 4   2             4   2        2 4   2      
     2a a a b b b c c  - 2a a b b c c  + 2a a a b b c c  + a a b b c c  -
       0 1 2 2 3 4 0 3     1 2 0 4 0 3     0 1 2 1 4 0 3    0 2 3 4 0 3  
     ------------------------------------------------------------------------
      2   3 3        2 4 2           3 3              4 2 2        
     a a a b c c  + a a b b c c  - 2a a b b b c c  + a a b b c c  -
      0 1 2 4 0 3    1 2 1 5 0 3     1 2 1 2 5 0 3    1 2 2 5 0 3  
     ------------------------------------------------------------------------
       2 4                   4               2 4 2           3 3            
     4a a b b b c c  + 4a a a b b b c c  - 2a a b b c c  + 4a a b b b c c  -
       1 2 0 3 5 0 3     0 1 2 1 3 5 0 3     0 2 3 5 0 3     1 2 0 4 5 0 3  
     ------------------------------------------------------------------------
         2 3                 3 2               2 2 2 2           4 2   2    
     6a a a b b b c c  + 2a a a b b b c c  + 3a a a b b c c  - 2a a b b c c 
       0 1 2 1 4 5 0 3     0 1 2 2 4 5 0 3     0 1 2 4 5 0 3     1 2 0 5 0 3
     ------------------------------------------------------------------------
           3 2   2           4     2         2 3     2         2 4 3      
     + 4a a a b b c c  - 2a a a b b c c  - 4a a a b b c c  + 2a a b c c  -
         0 1 2 1 5 0 3     0 1 2 2 5 0 3     0 1 2 4 5 0 3     0 1 5 0 3  
     ------------------------------------------------------------------------
      6                5                5                  4 2        
     a b b b c c  + a a b b b c c  + a a b b b c c  - a a a b b c c  +
      2 0 1 3 1 3    1 2 0 2 3 1 3    0 2 1 2 3 1 3    0 1 2 2 3 1 3  
     ------------------------------------------------------------------------
        5              2 4                  4                2 3 2        
     a a b b b c c  - a a b b b c c  - a a a b b b c c  + a a a b b c c  +
      1 2 0 1 4 1 3    1 2 0 2 4 1 3    0 1 2 1 2 4 1 3    0 1 2 2 4 1 3  
     ------------------------------------------------------------------------
        5              2 4                  4   2        2   3   2      
     a a b b b c c  - a a b b b c c  - a a a b b c c  + a a a b b c c  -
      0 2 0 3 4 1 3    0 2 2 3 4 1 3    0 1 2 0 4 1 3    0 1 2 2 4 1 3  
     ------------------------------------------------------------------------
      2 4              3 3                2 3                3 2 2        
     a a b b b c c  + a a b b b c c  + a a a b b b c c  - a a a b b c c  -
      1 2 0 1 5 1 3    1 2 0 2 5 1 3    0 1 2 1 2 5 1 3    0 1 2 2 5 1 3  
     ------------------------------------------------------------------------
           4              2 4               2   3            
     2a a a b b b c c  - a a b b b c c  + 3a a a b b b c c  +
       0 1 2 0 3 5 1 3    0 2 1 3 5 1 3     0 1 2 2 3 5 1 3  
     ------------------------------------------------------------------------
         2 3              2   3               2 2 2            
     3a a a b b b c c  + a a a b b b c c  - 4a a a b b b c c  +
       0 1 2 0 4 5 1 3    0 1 2 1 4 5 1 3     0 1 2 2 4 5 1 3  
     ------------------------------------------------------------------------
      3 3              3   2 2             3 2   2        2 2 2   2      
     a a b b b c c  - a a a b b c c  - 2a a a b b c c  - a a a b b c c  +
      0 2 3 4 5 1 3    0 1 2 4 5 1 3     0 1 2 0 5 1 3    0 1 2 1 5 1 3  
     ------------------------------------------------------------------------
       2 3     2         3   2   2         3 2     2         3 3 3      
     3a a a b b c c  - 2a a a b b c c  + 3a a a b b c c  - 2a a b c c  +
       0 1 2 2 5 1 3     0 1 2 3 5 1 3     0 1 2 4 5 1 3     0 1 5 1 3  
     ------------------------------------------------------------------------
        5                5 2          2 4                  4            
     a a b b b c c  - a a b b c c  - a a b b b c c  + a a a b b b c c  +
      1 2 0 1 3 2 3    0 2 1 3 2 3    1 2 0 2 3 2 3    0 1 2 1 2 3 2 3  
     ------------------------------------------------------------------------
         5   2         2 4   2        2 4                  4 2        
     2a a b b c c  - 2a a b b c c  - a a b b b c c  + a a a b b c c  +
       0 2 0 3 2 3     0 2 2 3 2 3    1 2 0 1 4 2 3    0 1 2 1 4 2 3  
     ------------------------------------------------------------------------
      3 3                2 3                   4               2 4          
     a a b b b c c  - a a a b b b c c  - 5a a a b b b c c  + 2a a b b b c c 
      1 2 0 2 4 2 3    0 1 2 1 2 4 2 3     0 1 2 0 3 4 2 3     0 2 1 3 4 2 3
     ------------------------------------------------------------------------
         2   3                 2 3   2         2   3   2        2 2 2   2    
     + 3a a a b b b c c  + 3a a a b b c c  - 2a a a b b c c  - a a a b b c c 
         0 1 2 2 3 4 2 3     0 1 2 0 4 2 3     0 1 2 1 4 2 3    0 1 2 2 4 2 3
     ------------------------------------------------------------------------
        3 3   2        3   2 3        3 3                2 3 2        
     - a a b b c c  + a a a b c c  + a a b b b c c  - a a a b b c c  -
        0 2 3 4 2 3    0 1 2 4 2 3    1 2 0 1 5 2 3    0 1 2 1 5 2 3  
     ------------------------------------------------------------------------
      4 2                3 2                 2 3            
     a a b b b c c  + a a a b b b c c  + 6a a a b b b c c  -
      1 2 0 2 5 2 3    0 1 2 1 2 5 2 3     0 1 2 0 3 5 2 3  
     ------------------------------------------------------------------------
       2   3               2 2 2               3 3 2        
     3a a a b b b c c  - 3a a a b b b c c  + 2a a b b c c  -
       0 1 2 1 3 5 2 3     0 1 2 2 3 5 2 3     0 2 3 5 2 3  
     ------------------------------------------------------------------------
         3 2               2 2 2               2 3              
     7a a a b b b c c  + 5a a a b b b c c  + 2a a a b b b c c  -
       0 1 2 0 4 5 2 3     0 1 2 1 4 5 2 3     0 1 2 2 4 5 2 3  
     ------------------------------------------------------------------------
      3   2               3 2   2             4     2         2 3     2      
     a a a b b b c c  - 2a a a b b c c  + 4a a a b b c c  - 3a a a b b c c  -
      0 1 2 3 4 5 2 3     0 1 2 4 5 2 3     0 1 2 0 5 2 3     0 1 2 1 5 2 3  
     ------------------------------------------------------------------------
      2 4   2         3 2     2        3 3   2        6 2   2       5       2
     a a b b c c  + 2a a a b b c c  + a a b b c c  + a b b c  - 2a a b b b c 
      0 1 2 5 2 3     0 1 2 3 5 2 3    0 1 4 5 2 3    2 0 3 3     0 2 0 2 3 3
     ------------------------------------------------------------------------
        2 4 2   2      5 2   2         4       2    2   3 2   2    2 4 2   2
     + a a b b c  - a a b b c  + 2a a a b b b c  - a a a b b c  + a a b b c 
        0 2 2 3 3    1 2 0 4 3     0 1 2 0 2 4 3    0 1 2 2 4 3    1 2 0 5 3
     ------------------------------------------------------------------------
           2 3       2    2 2 2 2   2     2 4       2     3 3       2  
     - 2a a a b b b c  + a a a b b c  + 2a a b b b c  - 2a a b b b c  -
         0 1 2 0 2 5 3    0 1 2 2 5 3     0 2 0 3 5 3     0 2 2 3 5 3  
     ------------------------------------------------------------------------
       2   3       2     3   2       2     2 2 2   2 2     3 2     2 2  
     2a a a b b b c  + 2a a a b b b c  + 2a a a b b c  - 2a a a b b c  +
       0 1 2 0 4 5 3     0 1 2 2 4 5 3     0 1 2 0 5 3     0 1 2 2 5 3  
     ------------------------------------------------------------------------
      4 2   2 2    4       2 2    4 2 3 2      5 2           2 4            
     a a b b c  - a a a b b c  + a a b c  - a a b b c c  + 2a a b b b c c  -
      0 2 3 5 3    0 1 2 4 5 3    0 1 5 3    1 2 1 3 0 4     1 2 1 2 3 0 4  
     ------------------------------------------------------------------------
      3 3 2             5   2          5   2             4   2      
     a a b b c c  + 2a a b b c c  + a a b b c c  - 3a a a b b c c  +
      1 2 2 3 0 4     1 2 0 3 0 4    0 2 1 3 0 4     0 1 2 2 3 0 4  
     ------------------------------------------------------------------------
      2 4 2           3 3              4 2 2           2 4            
     a a b b c c  - 2a a b b b c c  + a a b b c c  - 4a a b b b c c  +
      1 2 1 4 0 4     1 2 1 2 4 0 4    1 2 2 4 0 4     1 2 0 3 4 0 4  
     ------------------------------------------------------------------------
         2 3              2 4 2           3 3   2          2 3   2      
     4a a a b b b c c  - a a b b c c  + 2a a b b c c  - a a a b b c c  -
       0 1 2 2 3 4 0 4    0 2 3 4 0 4     1 2 0 4 0 4    0 1 2 1 4 0 4  
     ------------------------------------------------------------------------
        3 2   2        2   3   2        3 3 2           4 2            
     a a a b b c c  + a a a b b c c  - a a b b c c  + 2a a b b b c c  -
      0 1 2 2 4 0 4    0 1 2 3 4 0 4    1 2 1 5 0 4     1 2 1 2 5 0 4  
     ------------------------------------------------------------------------
      5   2           3 3                 2 3                 3 2            
     a a b b c c  + 4a a b b b c c  - 2a a a b b b c c  - 2a a a b b b c c  +
      1 2 2 5 0 4     1 2 0 3 5 0 4     0 1 2 1 3 5 0 4     0 1 2 2 3 5 0 4  
     ------------------------------------------------------------------------
       2   3 2           4 2                 3 2            
     4a a a b b c c  - 4a a b b b c c  + 4a a a b b b c c  -
       0 1 2 3 5 0 4     1 2 0 4 5 0 4     0 1 2 1 4 5 0 4  
     ------------------------------------------------------------------------
       2 2 2              2 3   2           5     2           4     2      
     6a a a b b b c c  + a a a b b c c  + 2a a b b c c  - 3a a a b b c c  +
       0 1 2 3 4 5 0 4    0 1 2 4 5 0 4     1 2 0 5 0 4     0 1 2 1 5 0 4  
     ------------------------------------------------------------------------
        5   2         2 3     2        2 4   2          5            
     a a b b c c  + 4a a a b b c c  - a a b b c c  + a a b b b c c  -
      0 1 2 5 0 4     0 1 2 3 5 0 4    0 1 4 5 0 4    1 2 0 1 3 1 4  
     ------------------------------------------------------------------------
      2 4                  4                2 3 2             5   2      
     a a b b b c c  - a a a b b b c c  + a a a b b c c  - 2a a b b c c  +
      1 2 0 2 3 1 4    0 1 2 1 2 3 1 4    0 1 2 2 3 1 4     0 2 0 3 1 4  
     ------------------------------------------------------------------------
       2 4   2        2 4              3 3                2 3            
     2a a b b c c  - a a b b b c c  + a a b b b c c  + a a a b b b c c  -
       0 2 2 3 1 4    1 2 0 1 4 1 4    1 2 0 2 4 1 4    0 1 2 1 2 4 1 4  
     ------------------------------------------------------------------------
        3 2 2               4               2   3                2 3   2    
     a a a b b c c  + 3a a a b b b c c  - 3a a a b b b c c  - a a a b b c c 
      0 1 2 2 4 1 4     0 1 2 0 3 4 1 4     0 1 2 2 3 4 1 4    0 1 2 0 4 1 4
     ------------------------------------------------------------------------
        2 2 2   2        3 3              4 2                3 2            
     + a a a b b c c  + a a b b b c c  - a a b b b c c  - a a a b b b c c  +
        0 1 2 2 4 1 4    1 2 0 1 5 1 4    1 2 0 2 5 1 4    0 1 2 1 2 5 1 4  
     ------------------------------------------------------------------------
        4   2             2 3              2   3              2 2 2          
     a a a b b c c  - 2a a a b b b c c  + a a a b b b c c  + a a a b b b c c 
      0 1 2 2 5 1 4     0 1 2 0 3 5 1 4    0 1 2 1 3 5 1 4    0 1 2 2 3 5 1 4
     ------------------------------------------------------------------------
         3 3 2            3 2              2 2 2            
     - 2a a b b c c  + a a a b b b c c  - a a a b b b c c  +
         0 2 3 5 1 4    0 1 2 0 4 5 1 4    0 1 2 1 4 5 1 4  
     ------------------------------------------------------------------------
       3   2              3 2   2          2 3     2        2 4   2      
     3a a a b b b c c  - a a a b b c c  + a a a b b c c  - a a b b c c  -
       0 1 2 3 4 5 1 4    0 1 2 4 5 1 4    0 1 2 1 5 1 4    0 1 2 5 1 4  
     ------------------------------------------------------------------------
       3 2     2        3 3   2        2 4                  4 2        
     2a a a b b c c  + a a b b c c  - a a b b b c c  + a a a b b c c  +
       0 1 2 3 5 1 4    0 1 4 5 1 4    1 2 0 1 3 2 4    0 1 2 1 3 2 4  
     ------------------------------------------------------------------------
      3 3                2 3              2 4   2        2   3   2      
     a a b b b c c  - a a a b b b c c  - a a b b c c  + a a a b b c c  +
      1 2 0 2 3 2 4    0 1 2 1 2 3 2 4    0 2 1 3 2 4    0 1 2 2 3 2 4  
     ------------------------------------------------------------------------
      3 3                2 3 2          4 2                3 2            
     a a b b b c c  - a a a b b c c  - a a b b b c c  + a a a b b b c c  +
      1 2 0 1 4 2 4    0 1 2 1 4 2 4    1 2 0 2 4 2 4    0 1 2 1 2 4 2 4  
     ------------------------------------------------------------------------
        2 3              2 2 2              3 3 2            3 2   2      
     a a a b b b c c  - a a a b b b c c  + a a b b c c  - a a a b b c c  +
      0 1 2 0 3 4 2 4    0 1 2 2 3 4 2 4    0 2 3 4 2 4    0 1 2 0 4 2 4  
     ------------------------------------------------------------------------
      2 2 2   2        3   2   2        4 2                3 2 2        
     a a a b b c c  - a a a b b c c  - a a b b b c c  + a a a b b c c  +
      0 1 2 1 4 2 4    0 1 2 3 4 2 4    1 2 0 1 5 2 4    0 1 2 1 5 2 4  
     ------------------------------------------------------------------------
      5                  4                   3 2              2 2 2          
     a a b b b c c  - a a a b b b c c  - 2a a a b b b c c  + a a a b b b c c 
      1 2 0 2 5 2 4    0 1 2 1 2 5 2 4     0 1 2 0 3 5 2 4    0 1 2 1 3 5 2 4
     ------------------------------------------------------------------------
        2 3                 3   2 2             4              
     + a a a b b b c c  - 2a a a b b c c  + 3a a a b b b c c  -
        0 1 2 2 3 5 2 4     0 1 2 3 5 2 4     0 1 2 0 4 5 2 4  
     ------------------------------------------------------------------------
       2 3                 3 2                   5   2         2 4   2      
     3a a a b b b c c  + 3a a a b b b c c  - 2a a b b c c  + 2a a b b c c  -
       0 1 2 1 4 5 2 4     0 1 2 3 4 5 2 4     0 1 0 5 2 4     0 1 1 5 2 4  
     ------------------------------------------------------------------------
       3 3   2           5 2            5                   4            
     2a a b b c c  - 2a a b b c c  + a a b b b c c  + 3a a a b b b c c  -
       0 1 3 5 2 4     1 2 0 3 3 4    0 2 0 1 3 3 4     0 1 2 0 2 3 3 4  
     ------------------------------------------------------------------------
      2 4              2   3 2           2 4 2              4            
     a a b b b c c  - a a a b b c c  + 2a a b b c c  - a a a b b b c c  -
      0 2 1 2 3 3 4    0 1 2 2 3 3 4     1 2 0 4 3 4    0 1 2 0 1 4 3 4  
     ------------------------------------------------------------------------
         2 3              2   3              2 2 2 2          2 4            
     3a a a b b b c c  + a a a b b b c c  + a a a b b c c  - a a b b b c c  +
       0 1 2 0 2 4 3 4    0 1 2 1 2 4 3 4    0 1 2 2 4 3 4    0 2 0 3 4 3 4  
     ------------------------------------------------------------------------
      3 3              2   3   2        3   2   2         3 3 2        
     a a b b b c c  + a a a b b c c  - a a a b b c c  - 2a a b b c c  +
      0 2 2 3 4 3 4    0 1 2 0 4 3 4    0 1 2 2 4 3 4     1 2 0 5 3 4  
     ------------------------------------------------------------------------
        2 3                 3 2              2 2 2              2 3   2      
     a a a b b b c c  + 3a a a b b b c c  - a a a b b b c c  - a a a b b c c 
      0 1 2 0 1 5 3 4     0 1 2 0 2 5 3 4    0 1 2 1 2 5 3 4    0 1 2 2 5 3 4
     ------------------------------------------------------------------------
         2   3              3 3              3   2            
     - 2a a a b b b c c  + a a b b b c c  + a a a b b b c c  +
         0 1 2 0 3 5 3 4    0 2 1 3 5 3 4    0 1 2 2 3 5 3 4  
     ------------------------------------------------------------------------
      2 2 2              3   2              4 2              4     2        
     a a a b b b c c  - a a a b b b c c  - a a b b b c c  + a a a b b c c  -
      0 1 2 0 4 5 3 4    0 1 2 1 4 5 3 4    0 2 3 4 5 3 4    0 1 2 4 5 3 4  
     ------------------------------------------------------------------------
       2 3     2        3 2     2        3 3   2        4 2   2      
     2a a a b b c c  + a a a b b c c  + a a b b c c  - a a b b c c  +
       0 1 2 0 5 3 4    0 1 2 1 5 3 4    0 1 2 5 3 4    0 1 4 5 3 4  
     ------------------------------------------------------------------------
      2 4 2   2        4       2      2 3       2    2   3       2  
     a a b b c  - a a a b b b c  - a a a b b b c  + a a a b b b c  +
      1 2 0 3 4    0 1 2 0 1 3 4    0 1 2 0 2 3 4    0 1 2 1 2 3 4  
     ------------------------------------------------------------------------
      2 4   2 2    3 3   2 2    3 3 2   2      2 3       2      3 2       2  
     a a b b c  - a a b b c  - a a b b c  + a a a b b b c  + a a a b b b c  -
      0 2 0 3 4    0 2 2 3 4    1 2 0 4 4    0 1 2 0 1 4 4    0 1 2 0 2 4 4  
     ------------------------------------------------------------------------
      2 2 2       2    2   3       2    3   2       2    4 2 2   2  
     a a a b b b c  - a a a b b b c  + a a a b b b c  + a a b b c  -
      0 1 2 1 2 4 4    0 1 2 0 3 4 4    0 1 2 2 3 4 4    1 2 0 5 4  
     ------------------------------------------------------------------------
        3 2       2      4         2    2 3         2     2 2 2       2  
     a a a b b b c  - a a a b b b c  + a a a b b b c  + 2a a a b b b c  -
      0 1 2 0 1 5 4    0 1 2 0 2 5 4    0 1 2 1 2 5 4     0 1 2 0 3 5 4  
     ------------------------------------------------------------------------
      3   2       2    3 2         2    4 2 2   2    2 3         2  
     a a a b b b c  - a a a b b b c  + a a b b c  - a a a b b b c  +
      0 1 2 1 3 5 4    0 1 2 2 3 5 4    0 2 3 5 4    0 1 2 0 4 5 4  
     ------------------------------------------------------------------------
      3 2         2    4           2    2 4   2 2    3 3   2 2    4 2   2 2  
     a a a b b b c  - a a a b b b c  + a a b b c  - a a b b c  + a a b b c  +
      0 1 2 1 4 5 4    0 1 2 3 4 5 4    0 1 0 5 4    0 1 1 5 4    0 1 3 5 4  
     ------------------------------------------------------------------------
      2 4 2           3 3              4 2 2           2 4   2      
     a a b b c c  - 2a a b b b c c  + a a b b c c  - 2a a b b c c  -
      1 2 1 3 0 5     1 2 1 2 3 0 5    1 2 2 3 0 5     1 2 0 3 0 5  
     ------------------------------------------------------------------------
           4   2           2 3   2         2 4 3        3 3 2        
     2a a a b b c c  + 4a a a b b c c  + 2a a b c c  - a a b b c c  +
       0 1 2 1 3 0 5     0 1 2 2 3 0 5     0 2 3 0 5    1 2 1 4 0 5  
     ------------------------------------------------------------------------
       4 2              5   2           3 3                 2 3            
     2a a b b b c c  - a a b b c c  + 4a a b b b c c  + 2a a a b b b c c  -
       1 2 1 2 4 0 5    1 2 2 4 0 5     1 2 0 3 4 0 5     0 1 2 1 3 4 0 5  
     ------------------------------------------------------------------------
         3 2               2   3 2           4 2   2           4     2      
     6a a a b b b c c  - 4a a a b b c c  - 2a a b b c c  + 2a a a b b c c  +
       0 1 2 2 3 4 0 5     0 1 2 3 4 0 5     1 2 0 4 0 5     0 1 2 2 4 0 5  
     ------------------------------------------------------------------------
       2 2 2   2        2 3   3        4 2 2           5              
     3a a a b b c c  - a a a b c c  + a a b b c c  - 2a a b b b c c  +
       0 1 2 3 4 0 5    0 1 2 4 0 5    1 2 1 5 0 5     1 2 1 2 5 0 5  
     ------------------------------------------------------------------------
      6 2           4 2                 4                 5              
     a b b c c  - 4a a b b b c c  + 4a a a b b b c c  + 4a a b b b c c  -
      1 2 5 0 5     1 2 0 3 5 0 5     0 1 2 2 3 5 0 5     1 2 0 4 5 0 5  
     ------------------------------------------------------------------------
         4                   5              2 4 2           6   2      
     2a a a b b b c c  - 2a a b b b c c  + a a b b c c  - 2a b b c c  +
       0 1 2 1 4 5 0 5     0 1 2 4 5 0 5    0 1 4 5 0 5     1 0 5 0 5  
     ------------------------------------------------------------------------
         5   2         2 4   2        2 4              3 3            
     2a a b b c c  - 2a a b b c c  - a a b b b c c  + a a b b b c c  +
       0 1 1 5 0 5     0 1 3 5 0 5    1 2 0 1 3 1 5    1 2 0 2 3 1 5  
     ------------------------------------------------------------------------
        2 3                3 2 2               4   2        2 4   2      
     a a a b b b c c  - a a a b b c c  + 4a a a b b c c  - a a b b c c  -
      0 1 2 1 2 3 1 5    0 1 2 2 3 1 5     0 1 2 0 3 1 5    0 2 1 3 1 5  
     ------------------------------------------------------------------------
       2   3   2        3 3              4 2                3 2            
     3a a a b b c c  + a a b b b c c  - a a b b b c c  - a a a b b b c c  +
       0 1 2 2 3 1 5    1 2 0 1 4 1 5    1 2 0 2 4 1 5    0 1 2 1 2 4 1 5  
     ------------------------------------------------------------------------
        4   2             2 3               2   3            
     a a a b b c c  - 7a a a b b b c c  + 2a a a b b b c c  +
      0 1 2 2 4 1 5     0 1 2 0 3 4 1 5     0 1 2 1 3 4 1 5  
     ------------------------------------------------------------------------
       2 2 2              3 3 2             3 2   2        2 2 2   2      
     5a a a b b b c c  + a a b b c c  + 3a a a b b c c  - a a a b b c c  -
       0 1 2 2 3 4 1 5    0 2 3 4 1 5     0 1 2 0 4 1 5    0 1 2 1 4 1 5  
     ------------------------------------------------------------------------
       2 3     2         3   2   2        3 2   3        4 2            
     2a a a b b c c  - 2a a a b b c c  + a a a b c c  - a a b b b c c  +
       0 1 2 2 4 1 5     0 1 2 3 4 1 5    0 1 2 4 1 5    1 2 0 1 5 1 5  
     ------------------------------------------------------------------------
      5                  4                  5 2             3 2            
     a a b b b c c  + a a a b b b c c  - a a b b c c  + 6a a a b b b c c  -
      1 2 0 2 5 1 5    0 1 2 1 2 5 1 5    0 1 2 5 1 5     0 1 2 0 3 5 1 5  
     ------------------------------------------------------------------------
       2 2 2               2 3                 3   2 2        
     3a a a b b b c c  - 3a a a b b b c c  + 2a a a b b c c  -
       0 1 2 1 3 5 1 5     0 1 2 2 3 5 1 5     0 1 2 3 5 1 5  
     ------------------------------------------------------------------------
         4                 2 3                 2 4            
     5a a a b b b c c  + 3a a a b b b c c  + 2a a b b b c c  -
       0 1 2 0 4 5 1 5     0 1 2 1 4 5 1 5     0 1 2 4 5 1 5  
     ------------------------------------------------------------------------
      3 2                3 3 2             5   2         2 4   2      
     a a a b b b c c  - a a b b c c  + 2a a b b c c  - 2a a b b c c  +
      0 1 2 3 4 5 1 5    0 1 4 5 1 5     0 1 0 5 1 5     0 1 1 5 1 5  
     ------------------------------------------------------------------------
       3 3   2        3 3                2 3 2          4 2            
     2a a b b c c  + a a b b b c c  - a a a b b c c  - a a b b b c c  +
       0 1 3 5 1 5    1 2 0 1 3 2 5    0 1 2 1 3 2 5    1 2 0 2 3 2 5  
     ------------------------------------------------------------------------
        3 2                 2 3   2         2   3   2        2 2 2   2      
     a a a b b b c c  - 2a a a b b c c  + 3a a a b b c c  - a a a b b c c  -
      0 1 2 1 2 3 2 5     0 1 2 0 3 2 5     0 1 2 1 3 2 5    0 1 2 2 3 2 5  
     ------------------------------------------------------------------------
       3 3 3        4 2                3 2 2          5              
     2a a b c c  - a a b b b c c  + a a a b b c c  + a a b b b c c  -
       0 2 3 2 5    1 2 0 1 4 2 5    0 1 2 1 4 2 5    1 2 0 2 4 2 5  
     ------------------------------------------------------------------------
        4                   3 2               2 2 2            
     a a a b b b c c  + 3a a a b b b c c  - 4a a a b b b c c  +
      0 1 2 1 2 4 2 5     0 1 2 0 3 4 2 5     0 1 2 1 3 4 2 5  
     ------------------------------------------------------------------------
      2 3                 3   2 2            4     2        2 3     2      
     a a a b b b c c  + 3a a a b b c c  - a a a b b c c  + a a a b b c c  -
      0 1 2 2 3 4 2 5     0 1 2 3 4 2 5    0 1 2 0 4 2 5    0 1 2 1 4 2 5  
     ------------------------------------------------------------------------
      3 2     2        5                  4   2          6            
     a a a b b c c  + a a b b b c c  - a a a b b c c  - a b b b c c  +
      0 1 2 3 4 2 5    1 2 0 1 5 2 5    0 1 2 1 5 2 5    1 0 2 5 2 5  
     ------------------------------------------------------------------------
        5                 4                 2 3                2 4          
     a a b b b c c  - 2a a a b b b c c  + 3a a a b b b c c  - a a b b b c c 
      0 1 1 2 5 2 5     0 1 2 0 3 5 2 5     0 1 2 1 3 5 2 5    0 1 2 3 5 2 5
     ------------------------------------------------------------------------
         3 2   2            5              2 4              3 3            
     - 2a a a b b c c  + a a b b b c c  - a a b b b c c  + a a b b b c c  +
         0 1 2 3 5 2 5    0 1 0 4 5 2 5    0 1 1 4 5 2 5    0 1 3 4 5 2 5  
     ------------------------------------------------------------------------
       2 4 2               4              2 4 2             2 3            
     2a a b b c c  - 2a a a b b b c c  + a a b b c c  - 2a a a b b b c c  +
       1 2 0 3 3 5     0 1 2 0 1 3 3 5    0 2 1 3 3 5     0 1 2 0 2 3 3 5  
     ------------------------------------------------------------------------
      2 2 2 2           2 4   2         3 3   2         3 3 2        
     a a a b b c c  - 2a a b b c c  + 2a a b b c c  - 2a a b b c c  +
      0 1 2 2 3 3 5     0 2 0 3 3 5     0 2 2 3 3 5     1 2 0 4 3 5  
     ------------------------------------------------------------------------
         2 3              2   3 2             3 2              2 3   2      
     2a a a b b b c c  - a a a b b c c  + 2a a a b b b c c  - a a a b b c c 
       0 1 2 0 1 4 3 5    0 1 2 1 4 3 5     0 1 2 0 2 4 3 5    0 1 2 2 4 3 5
     ------------------------------------------------------------------------
         2   3               3 3               3   2            
     + 6a a a b b b c c  - 2a a b b b c c  - 4a a a b b b c c  -
         0 1 2 0 3 4 3 5     0 2 1 3 4 3 5     0 1 2 2 3 4 3 5  
     ------------------------------------------------------------------------
       2 2 2   2         3   2   2         3 2     2        4 2   2      
     4a a a b b c c  + 2a a a b b c c  + 2a a a b b c c  + a a b b c c  -
       0 1 2 0 4 3 5     0 1 2 1 4 3 5     0 1 2 2 4 3 5    0 2 3 4 3 5  
     ------------------------------------------------------------------------
      4     3         4 2 2             3 2              2 2 2 2        
     a a a b c c  + 2a a b b c c  - 2a a a b b b c c  + a a a b b c c  -
      0 1 2 4 3 5     1 2 0 5 3 5     0 1 2 0 1 5 3 5    0 1 2 1 5 3 5  
     ------------------------------------------------------------------------
         4                2 4 2           2 2 2               3   2          
     2a a a b b b c c  + a a b b c c  - 4a a a b b b c c  + 2a a a b b b c c 
       0 1 2 0 2 5 3 5    0 1 2 5 3 5     0 1 2 0 3 5 3 5     0 1 2 1 3 5 3 5
     ------------------------------------------------------------------------
         3 2                 4 2 2           2 3              
     + 2a a a b b b c c  - 2a a b b c c  + 6a a a b b b c c  -
         0 1 2 2 3 5 3 5     0 2 3 5 3 5     0 1 2 0 4 5 3 5  
     ------------------------------------------------------------------------
       3 2                 3 3               4                  4 2 2        
     4a a a b b b c c  - 2a a b b b c c  + 2a a a b b b c c  + a a b b c c  -
       0 1 2 1 4 5 3 5     0 1 2 4 5 3 5     0 1 2 3 4 5 3 5    0 1 4 5 3 5  
     ------------------------------------------------------------------------
       2 4   2         3 3   2         4 2   2         3 3 2        
     2a a b b c c  + 2a a b b c c  - 2a a b b c c  - 2a a b b c c  +
       0 1 0 5 3 5     0 1 1 5 3 5     0 1 3 5 3 5     1 2 0 3 4 5  
     ------------------------------------------------------------------------
         2 3              2   3 2            3 2              2 2 2          
     3a a a b b b c c  - a a a b b c c  + a a a b b b c c  - a a a b b b c c 
       0 1 2 0 1 3 4 5    0 1 2 1 3 4 5    0 1 2 0 2 3 4 5    0 1 2 1 2 3 4 5
     ------------------------------------------------------------------------
         2   3   2        3 3   2        3   2   2         4 2 2        
     - 2a a a b b c c  + a a b b c c  + a a a b b c c  + 2a a b b c c  -
         0 1 2 0 3 4 5    0 2 1 3 4 5    0 1 2 2 3 4 5     1 2 0 4 4 5  
     ------------------------------------------------------------------------
         3 2              2 2 2 2            4                2 3            
     3a a a b b b c c  + a a a b b c c  - a a a b b b c c  + a a a b b b c c 
       0 1 2 0 1 4 4 5    0 1 2 1 4 4 5    0 1 2 0 2 4 4 5    0 1 2 1 2 4 4 5
     ------------------------------------------------------------------------
        2 2 2              3 2                4 2 2          2 3     2      
     + a a a b b b c c  - a a a b b b c c  - a a b b c c  + a a a b b c c  -
        0 1 2 0 3 4 4 5    0 1 2 2 3 4 4 5    0 2 3 4 4 5    0 1 2 0 4 4 5  
     ------------------------------------------------------------------------
      3 2     2        4       2         5   2             4              
     a a a b b c c  + a a a b b c c  - 2a a b b c c  + 3a a a b b b c c  -
      0 1 2 1 4 4 5    0 1 2 3 4 4 5     1 2 0 5 4 5     0 1 2 0 1 5 4 5  
     ------------------------------------------------------------------------
      2 3   2            5              2 4               2 3              
     a a a b b c c  + a a b b b c c  - a a b b b c c  - 2a a a b b b c c  +
      0 1 2 1 5 4 5    0 1 0 2 5 4 5    0 1 1 2 5 4 5     0 1 2 0 3 5 4 5  
     ------------------------------------------------------------------------
      3 2                3 3              2 4              3 3            
     a a a b b b c c  + a a b b b c c  - a a b b b c c  + a a b b b c c  -
      0 1 2 1 3 5 4 5    0 1 2 3 5 4 5    0 1 0 4 5 4 5    0 1 1 4 5 4 5  
     ------------------------------------------------------------------------
      4 2              4 2 2   2       3 2       2    2 2 2 2   2  
     a a b b b c c  + a a b b c  - 2a a a b b b c  + a a a b b c  +
      0 1 3 4 5 4 5    1 2 0 3 5     0 1 2 0 1 3 5    0 1 2 1 3 5  
     ------------------------------------------------------------------------
       2 2 2   2 2     3   2   2 2    4 2 3 2    5   2   2       4         2
     2a a a b b c  - 2a a a b b c  + a a b c  - a a b b c  + 2a a a b b b c 
       0 1 2 0 3 5     0 1 2 1 3 5    0 2 3 5    1 2 0 4 5     0 1 2 0 1 4 5
     ------------------------------------------------------------------------
        2 3   2   2     2 3         2     3 2         2    4     2   2  
     - a a a b b c  - 2a a a b b b c  + 2a a a b b b c  - a a a b b c  +
        0 1 2 1 4 5     0 1 2 0 3 4 5     0 1 2 1 3 4 5    0 1 2 3 4 5  
     ------------------------------------------------------------------------
      6 2   2       5       2    2 4 2   2     2 4       2     3 3       2  
     a b b c  - 2a a b b b c  + a a b b c  + 2a a b b b c  - 2a a b b b c  +
      1 0 5 5     0 1 0 1 5 5    0 1 1 5 5     0 1 0 3 5 5     0 1 1 3 5 5  
     ------------------------------------------------------------------------
      4 2 2   2    6 3           5 2           2 4   2        3 3 3      
     a a b b c  - a b c c  + 3a a b b c c  - 3a a b b c c  + a a b c c  +
      0 1 3 5 5    2 1 0 6     1 2 1 2 0 6     1 2 1 2 0 6    1 2 2 0 6  
     ------------------------------------------------------------------------
       6                 5                 5                   4 2        
     3a b b b c c  - 3a a b b b c c  - 3a a b b b c c  + 3a a a b b c c  -
       2 0 1 3 0 6     1 2 0 2 3 0 6     0 2 1 2 3 0 6     0 1 2 2 3 0 6  
     ------------------------------------------------------------------------
         5                 5 2           2 4                   4            
     3a a b b b c c  + 3a a b b c c  + 3a a b b b c c  - 3a a a b b b c c  -
       1 2 0 1 4 0 6     0 2 1 4 0 6     1 2 0 2 4 0 6     0 1 2 1 2 4 0 6  
     ------------------------------------------------------------------------
         5               2 4                   4   2         2 4   2      
     3a a b b b c c  + 3a a b b b c c  + 3a a a b b c c  - 3a a b b c c  +
       0 2 0 3 4 0 6     0 2 2 3 4 0 6     0 1 2 0 4 0 6     0 2 1 4 0 6  
     ------------------------------------------------------------------------
      3 3 3         2 4                   4 2           3 3            
     a a b c c  + 3a a b b b c c  - 6a a a b b c c  - 3a a b b b c c  +
      0 2 4 0 6     1 2 0 1 5 0 6     0 1 2 1 5 0 6     1 2 0 2 5 0 6  
     ------------------------------------------------------------------------
         2 3                 3 2 2               4            
     9a a a b b b c c  - 3a a a b b c c  + 6a a a b b b c c  +
       0 1 2 1 2 5 0 6     0 1 2 2 5 0 6     0 1 2 0 3 5 0 6  
     ------------------------------------------------------------------------
       2 4               2   3                 2 3            
     3a a b b b c c  - 9a a a b b b c c  - 9a a a b b b c c  +
       0 2 1 3 5 0 6     0 1 2 2 3 5 0 6     0 1 2 0 4 5 0 6  
     ------------------------------------------------------------------------
       2   3               3 3               3   2 2             3 2   2    
     9a a a b b b c c  - 3a a b b b c c  - 3a a a b b c c  + 6a a a b b c c 
       0 1 2 1 4 5 0 6     0 2 3 4 5 0 6     0 1 2 4 5 0 6     0 1 2 0 5 0 6
     ------------------------------------------------------------------------
         2 2 2   2         2 3     2         3   2   2         3 2     2    
     - 9a a a b b c c  + 3a a a b b c c  + 6a a a b b c c  + 3a a a b b c c 
         0 1 2 1 5 0 6     0 1 2 2 5 0 6     0 1 2 3 5 0 6     0 1 2 4 5 0 6
     ------------------------------------------------------------------------
         3 3 3        6   2           5                5 2        
     - 2a a b c c  + a b b c c  - 2a a b b b c c  - a a b b c c  +
         0 1 5 0 6    2 0 1 1 6     1 2 0 1 2 1 6    0 2 1 2 1 6  
     ------------------------------------------------------------------------
      2 4   2             4   2          2 3 3         6 2        
     a a b b c c  + 2a a a b b c c  - a a a b c c  - 2a b b c c  +
      1 2 0 2 1 6     0 1 2 1 2 1 6    0 1 2 2 1 6     2 0 3 1 6  
     ------------------------------------------------------------------------
         5               2 4 2             5 2             5            
     4a a b b b c c  - 2a a b b c c  + 2a a b b c c  - 2a a b b b c c  -
       0 2 0 2 3 1 6     0 2 2 3 1 6     1 2 0 4 1 6     0 2 0 1 4 1 6  
     ------------------------------------------------------------------------
           4               2 4              2 4   2        3 3   2      
     2a a a b b b c c  + 2a a b b b c c  + a a b b c c  - a a b b c c  -
       0 1 2 0 2 4 1 6     0 2 1 2 4 1 6    0 2 0 4 1 6    0 2 2 4 1 6  
     ------------------------------------------------------------------------
       2 4 2               4              2 4 2           2   3            
     2a a b b c c  + 4a a a b b b c c  + a a b b c c  - 6a a a b b b c c  +
       1 2 0 5 1 6     0 1 2 0 1 5 1 6    0 2 1 5 1 6     0 1 2 1 2 5 1 6  
     ------------------------------------------------------------------------
       2 2 2 2           2 4               3 3               3 3            
     3a a a b b c c  - 4a a b b b c c  + 4a a b b b c c  - 2a a b b b c c  +
       0 1 2 2 5 1 6     0 2 0 3 5 1 6     0 2 2 3 5 1 6     0 2 1 4 5 1 6  
     ------------------------------------------------------------------------
       3   2              4 2 2           3   2   2         3 2     2      
     2a a a b b b c c  + a a b b c c  + 4a a a b b c c  - 4a a a b b c c  -
       0 1 2 2 4 5 1 6    0 2 4 5 1 6     0 1 2 1 5 1 6     0 1 2 2 5 1 6  
     ------------------------------------------------------------------------
       4 2   2         4       2         4 2 3          5   2      
     2a a b b c c  - 2a a a b b c c  + 2a a b c c  - a a b b c c  +
       0 2 3 5 1 6     0 1 2 4 5 1 6     0 1 5 1 6    1 2 0 1 2 6  
     ------------------------------------------------------------------------
        5 3         2 4                   4 2          3 3   2      
     a a b c c  + 2a a b b b c c  - 2a a a b b c c  - a a b b c c  +
      0 2 1 2 6     1 2 0 1 2 2 6     0 1 2 1 2 2 6    1 2 0 2 2 6  
     ------------------------------------------------------------------------
        2 3   2           5 2             5                  4            
     a a a b b c c  + 2a a b b c c  - 3a a b b b c c  - a a a b b b c c  +
      0 1 2 1 2 2 6     1 2 0 3 2 6     0 2 0 1 3 2 6    0 1 2 0 2 3 2 6  
     ------------------------------------------------------------------------
       2 4              2   3 2           2 4 2               4            
     3a a b b b c c  - a a a b b c c  - 2a a b b c c  + 5a a a b b b c c  -
       0 2 1 2 3 2 6    0 1 2 2 3 2 6     1 2 0 4 2 6     0 1 2 0 1 4 2 6  
     ------------------------------------------------------------------------
       2 4 2            2 3              2   3               2 4            
     3a a b b c c  - a a a b b b c c  + a a a b b b c c  + 3a a b b b c c  -
       0 2 1 4 2 6    0 1 2 0 2 4 2 6    0 1 2 1 2 4 2 6     0 2 0 3 4 2 6  
     ------------------------------------------------------------------------
       3 3               2   3   2         3 3   2        3   2   2      
     3a a b b b c c  - 4a a a b b c c  + 3a a b b c c  + a a a b b c c  -
       0 2 2 3 4 2 6     0 1 2 0 4 2 6     0 2 1 4 2 6    0 1 2 2 4 2 6  
     ------------------------------------------------------------------------
      4 2 3         3 3 2             2 3               2   3 2        
     a a b c c  + 2a a b b c c  - 7a a a b b b c c  + 5a a a b b c c  +
      0 2 4 2 6     1 2 0 5 2 6     0 1 2 0 1 5 2 6     0 1 2 1 5 2 6  
     ------------------------------------------------------------------------
         3 2               2 2 2               2   3            
     3a a a b b b c c  - 3a a a b b b c c  - 2a a a b b b c c  -
       0 1 2 0 2 5 2 6     0 1 2 1 2 5 2 6     0 1 2 0 3 5 2 6  
     ------------------------------------------------------------------------
       3 3               3   2               2 2 2            
     3a a b b b c c  + 5a a a b b b c c  + 9a a a b b b c c  -
       0 2 1 3 5 2 6     0 1 2 2 3 5 2 6     0 1 2 0 4 5 2 6  
     ------------------------------------------------------------------------
       3   2               3 2                 4 2            
     7a a a b b b c c  - 2a a a b b b c c  + 3a a b b b c c  +
       0 1 2 1 4 5 2 6     0 1 2 2 4 5 2 6     0 2 3 4 5 2 6  
     ------------------------------------------------------------------------
       4     2           2 3     2         3 2     2        3 3   2      
     2a a a b b c c  - 6a a a b b c c  + 5a a a b b c c  + a a b b c c  -
       0 1 2 4 5 2 6     0 1 2 0 5 2 6     0 1 2 1 5 2 6    0 1 2 5 2 6  
     ------------------------------------------------------------------------
       4       2        4 2   2        6 2            5 2        
     4a a a b b c c  - a a b b c c  - a b b c c  + a a b b c c  +
       0 1 2 3 5 2 6    0 1 4 5 2 6    2 0 1 3 6    1 2 0 2 3 6  
     ------------------------------------------------------------------------
         5                   4   2        2 4   2        2   3 3      
     2a a b b b c c  - 2a a a b b c c  - a a b b c c  + a a a b c c  +
       0 2 0 1 2 3 6     0 1 2 0 2 3 6    0 2 1 2 3 6    0 1 2 2 3 6  
     ------------------------------------------------------------------------
        5 2           2 4              3 3 2               4 2        
     a a b b c c  - 2a a b b b c c  + a a b b c c  - 2a a a b b c c  -
      0 2 0 4 3 6     0 2 0 2 4 3 6    0 2 2 4 3 6     0 1 2 0 5 3 6  
     ------------------------------------------------------------------------
       2 4               2   3               3 3               3   2 2      
     2a a b b b c c  + 6a a a b b b c c  + 2a a b b b c c  - 4a a a b b c c 
       0 2 0 1 5 3 6     0 1 2 0 2 5 3 6     0 2 1 2 5 3 6     0 1 2 2 5 3 6
     ------------------------------------------------------------------------
         3 3               4 2               3   2   2        4 2   2      
     + 2a a b b b c c  - 2a a b b b c c  - 4a a a b b c c  - a a b b c c  +
         0 2 0 4 5 3 6     0 2 2 4 5 3 6     0 1 2 0 5 3 6    0 2 1 5 3 6  
     ------------------------------------------------------------------------
       4       2        5     2         5   3          5 2        
     5a a a b b c c  + a a b b c c  - 2a a b c c  + a a b b c c  -
       0 1 2 2 5 3 6    0 2 4 5 3 6     0 1 5 3 6    1 2 0 1 4 6  
     ------------------------------------------------------------------------
        5   2        2 4 2          2 4 2            2 3   2      
     a a b b c c  - a a b b c c  + a a b b c c  + a a a b b c c  -
      0 2 0 1 4 6    1 2 0 2 4 6    0 2 1 2 4 6    0 1 2 0 2 4 6  
     ------------------------------------------------------------------------
      2   3   2           5 2           2 4               3 3 2        
     a a a b b c c  + 2a a b b c c  - 4a a b b b c c  + 2a a b b c c  -
      0 1 2 1 2 4 6     0 2 0 3 4 6     0 2 0 2 3 4 6     0 2 2 3 4 6  
     ------------------------------------------------------------------------
           4 2           2 4               2   3               3 3          
     3a a a b b c c  + 2a a b b b c c  + 4a a a b b b c c  - 2a a b b b c c 
       0 1 2 0 4 4 6     0 2 0 1 4 4 6     0 1 2 0 2 4 4 6     0 2 1 2 4 4 6
     ------------------------------------------------------------------------
        3   2 2          3 3   2        4 2   2           2 3 2        
     - a a a b b c c  - a a b b c c  + a a b b c c  + 4a a a b b c c  -
        0 1 2 2 4 4 6    0 2 0 4 4 6    0 2 2 4 4 6     0 1 2 0 5 4 6  
     ------------------------------------------------------------------------
       2   3              3 3 2           2 2 2               3   2          
     2a a a b b b c c  - a a b b c c  - 6a a a b b b c c  + 4a a a b b b c c 
       0 1 2 0 1 5 4 6    0 2 1 5 4 6     0 1 2 0 2 5 4 6     0 1 2 1 2 5 4 6
     ------------------------------------------------------------------------
        3 2   2           3 3               4 2               3   2          
     + a a a b b c c  + 4a a b b b c c  - 4a a b b b c c  - 2a a a b b b c c 
        0 1 2 2 5 4 6     0 2 0 3 5 4 6     0 2 2 3 5 4 6     0 1 2 0 4 5 4 6
     ------------------------------------------------------------------------
         4 2              5   2           3 2     2         4       2      
     + 2a a b b b c c  - a a b b c c  + 4a a a b b c c  - 3a a a b b c c  -
         0 2 1 4 5 4 6    0 2 4 5 4 6     0 1 2 0 5 4 6     0 1 2 1 5 4 6  
     ------------------------------------------------------------------------
      4 2   2         5     2        5     2        2 4 2        
     a a b b c c  + 2a a b b c c  + a a b b c c  - a a b b c c  +
      0 1 2 5 4 6     0 2 3 5 4 6    0 1 4 5 4 6    1 2 0 1 5 6  
     ------------------------------------------------------------------------
           4   2        2 4 3        3 3 2             2 3            
     2a a a b b c c  - a a b c c  + a a b b c c  - 2a a a b b b c c  +
       0 1 2 0 1 5 6    0 2 1 5 6    1 2 0 2 5 6     0 1 2 0 1 2 5 6  
     ------------------------------------------------------------------------
      2   3 2               4 2           2 4               2   3            
     a a a b b c c  - 4a a a b b c c  + 3a a b b b c c  + 5a a a b b b c c  -
      0 1 2 1 2 5 6     0 1 2 0 3 5 6     0 2 0 1 3 5 6     0 1 2 0 2 3 5 6  
     ------------------------------------------------------------------------
       3 3              3   2 2             2 3 2           2   3            
     3a a b b b c c  - a a a b b c c  + 5a a a b b c c  - 7a a a b b b c c  +
       0 2 1 2 3 5 6    0 1 2 2 3 5 6     0 1 2 0 4 5 6     0 1 2 0 1 4 5 6  
     ------------------------------------------------------------------------
       3 3 2           2 2 2              3   2              3 2   2        
     3a a b b c c  - 3a a a b b b c c  + a a a b b b c c  + a a a b b c c  -
       0 2 1 4 5 6     0 1 2 0 2 4 5 6    0 1 2 1 2 4 5 6    0 1 2 2 4 5 6  
     ------------------------------------------------------------------------
       3 3               4 2               3   2   2         4 2   2      
     3a a b b b c c  + 3a a b b b c c  + 5a a a b b c c  - 3a a b b c c  -
       0 2 0 3 4 5 6     0 2 2 3 4 5 6     0 1 2 0 4 5 6     0 2 1 4 5 6  
     ------------------------------------------------------------------------
       4       2        5   3           3 2 2           2 2 2            
     2a a a b b c c  + a a b c c  - 6a a a b b c c  + 9a a a b b b c c  -
       0 1 2 2 4 5 6    0 2 4 5 6     0 1 2 0 5 5 6     0 1 2 0 1 5 5 6  
     ------------------------------------------------------------------------
       3   2 2           2 3                3 2                3 3 2        
     4a a a b b c c  + 3a a a b b b c c  - a a a b b b c c  - a a b b c c  -
       0 1 2 1 5 5 6     0 1 2 0 2 5 5 6    0 1 2 1 2 5 5 6    0 1 2 5 5 6  
     ------------------------------------------------------------------------
       3   2               4 2              4                
     2a a a b b b c c  + 3a a b b b c c  - a a a b b b c c  -
       0 1 2 0 3 5 5 6     0 2 1 3 5 5 6    0 1 2 2 3 5 5 6  
     ------------------------------------------------------------------------
       3 2                 4                   4 2            
     7a a a b b b c c  + 5a a a b b b c c  + 2a a b b b c c  -
       0 1 2 0 4 5 5 6     0 1 2 1 4 5 5 6     0 1 2 4 5 5 6  
     ------------------------------------------------------------------------
       5                5   2           3 3   2         4 2   2      
     3a a b b b c c  - a a b b c c  + 2a a b b c c  - 2a a b b c c  +
       0 2 3 4 5 5 6    0 1 4 5 5 6     0 1 0 5 5 6     0 1 1 5 5 6  
     ------------------------------------------------------------------------
       5     2        6 3 2       5 2   2     2 4   2 2    3 3 3 2  
     2a a b b c c  + a b c  - 3a a b b c  + 3a a b b c  - a a b c  +
       0 1 3 5 5 6    2 0 6     0 2 0 2 6     0 2 0 2 6    0 2 2 6  
     ------------------------------------------------------------------------
       2 4 2   2     3 3       2     4 2 2   2     4 2   2 2     5     2 2  
     3a a b b c  - 6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  +
       0 2 0 5 6     0 2 0 2 5 6     0 2 2 5 6     0 2 0 5 6     0 2 2 5 6  
     ------------------------------------------------------------------------
      6 3 2      5 3         2 4 2           3 3   2        4 2 3      
     a b c  + a a b c c  - 3a a b b c c  + 3a a b b c c  - a a b c c  -
      0 5 6    1 2 1 0 7     1 2 1 2 0 7     1 2 1 2 0 7    1 2 2 0 7  
     ------------------------------------------------------------------------
         5                5 2           2 4                   4            
     3a a b b b c c  - a a b b c c  + 3a a b b b c c  + 5a a a b b b c c  -
       1 2 0 1 3 0 7    0 2 1 3 0 7     1 2 0 2 3 0 7     0 1 2 1 2 3 0 7  
     ------------------------------------------------------------------------
         2 3 2             5   2         2 4   2         2 4            
     4a a a b b c c  + 2a a b b c c  - 2a a b b c c  + 3a a b b b c c  -
       0 1 2 2 3 0 7     0 2 0 3 0 7     0 2 2 3 0 7     1 2 0 1 4 0 7  
     ------------------------------------------------------------------------
           4 2           3 3                2 3                3 2 2        
     2a a a b b c c  - 3a a b b b c c  + a a a b b b c c  + a a a b b c c  -
       0 1 2 1 4 0 7     1 2 0 2 4 0 7    0 1 2 1 2 4 0 7    0 1 2 2 4 0 7  
     ------------------------------------------------------------------------
          4               2 4              2   3                2 3   2      
     a a a b b b c c  + 2a a b b b c c  - a a a b b b c c  - a a a b b c c  +
      0 1 2 0 3 4 0 7     0 2 1 3 4 0 7    0 1 2 2 3 4 0 7    0 1 2 0 4 0 7  
     ------------------------------------------------------------------------
      2   3   2        3 3   2         3 3                 2 3 2        
     a a a b b c c  - a a b b c c  - 3a a b b b c c  + 5a a a b b c c  +
      0 1 2 1 4 0 7    0 2 3 4 0 7     1 2 0 1 5 0 7     0 1 2 1 5 0 7  
     ------------------------------------------------------------------------
       4 2                 3 2                 4   2        
     3a a b b b c c  - 7a a a b b b c c  + 2a a a b b c c  -
       1 2 0 2 5 0 7     0 1 2 1 2 5 0 7     0 1 2 2 5 0 7  
     ------------------------------------------------------------------------
         2 3               2   3               2 2 2            
     2a a a b b b c c  - 7a a a b b b c c  + 9a a a b b b c c  +
       0 1 2 0 3 5 0 7     0 1 2 1 3 5 0 7     0 1 2 2 3 5 0 7  
     ------------------------------------------------------------------------
       3 3 2             3 2               2 2 2            
     2a a b b c c  + 5a a a b b b c c  - 3a a a b b b c c  -
       0 2 3 5 0 7     0 1 2 0 4 5 0 7     0 1 2 1 4 5 0 7  
     ------------------------------------------------------------------------
       2 3                 3   2                 4     2      
     2a a a b b b c c  + 3a a a b b b c c  - 4a a a b b c c  +
       0 1 2 2 4 5 0 7     0 1 2 3 4 5 0 7     0 1 2 0 5 0 7  
     ------------------------------------------------------------------------
       2 3     2        2 4   2         3 2     2        3 3   2      
     5a a a b b c c  - a a b b c c  - 6a a a b b c c  + a a b b c c  -
       0 1 2 1 5 0 7    0 1 2 5 0 7     0 1 2 3 5 0 7    0 1 4 5 0 7  
     ------------------------------------------------------------------------
        5   2         2 4                  4 2          3 3   2      
     a a b b c c  + 2a a b b b c c  + a a a b b c c  - a a b b c c  -
      1 2 0 1 1 7     1 2 0 1 2 1 7    0 1 2 1 2 1 7    1 2 0 2 1 7  
     ------------------------------------------------------------------------
         2 3   2          3 2 3           5 2            5            
     2a a a b b c c  + a a a b c c  + 2a a b b c c  + a a b b b c c  -
       0 1 2 1 2 1 7    0 1 2 2 1 7     1 2 0 3 1 7    0 2 0 1 3 1 7  
     ------------------------------------------------------------------------
           4              2 4               2   3 2           2 4 2        
     5a a a b b b c c  - a a b b b c c  + 3a a a b b c c  - 2a a b b c c  +
       0 1 2 0 2 3 1 7    0 2 1 2 3 1 7     0 1 2 2 3 1 7     1 2 0 4 1 7  
     ------------------------------------------------------------------------
          4                 2 3              2   3              2 2 2 2      
     a a a b b b c c  + 3a a a b b b c c  - a a a b b b c c  - a a a b b c c 
      0 1 2 0 1 4 1 7     0 1 2 0 2 4 1 7    0 1 2 1 2 4 1 7    0 1 2 2 4 1 7
     ------------------------------------------------------------------------
        2 4              3 3               3 3 2             2 3            
     - a a b b b c c  + a a b b b c c  + 2a a b b c c  - 3a a a b b b c c  -
        0 2 0 3 4 1 7    0 2 2 3 4 1 7     1 2 0 5 1 7     0 1 2 0 1 5 1 7  
     ------------------------------------------------------------------------
      2   3 2            3 2               2 2 2               2 3   2      
     a a a b b c c  - a a a b b b c c  + 5a a a b b b c c  - 2a a a b b c c 
      0 1 2 1 5 1 7    0 1 2 0 2 5 1 7     0 1 2 1 2 5 1 7     0 1 2 2 5 1 7
     ------------------------------------------------------------------------
         2   3              3 3               3   2            
     + 6a a a b b b c c  + a a b b b c c  - 7a a a b b b c c  -
         0 1 2 0 3 5 1 7    0 2 1 3 5 1 7     0 1 2 2 3 5 1 7  
     ------------------------------------------------------------------------
       2 2 2              3   2               3 2              
     3a a a b b b c c  + a a a b b b c c  + 2a a a b b b c c  -
       0 1 2 0 4 5 1 7    0 1 2 1 4 5 1 7     0 1 2 2 4 5 1 7  
     ------------------------------------------------------------------------
      4 2               2 3     2         3 2     2        3 3   2      
     a a b b b c c  + 2a a a b b c c  - 3a a a b b c c  + a a b b c c  +
      0 2 3 4 5 1 7     0 1 2 0 5 1 7     0 1 2 1 5 1 7    0 1 2 5 1 7  
     ------------------------------------------------------------------------
       4       2        4 2   2        2 4   2            4 3      
     4a a a b b c c  - a a b b c c  + a a b b c c  - a a a b c c  -
       0 1 2 3 5 1 7    0 1 4 5 1 7    1 2 0 1 2 7    0 1 2 1 2 7  
     ------------------------------------------------------------------------
       3 3                 2 3 2          4 2   2          3 2   2      
     2a a b b b c c  + 2a a a b b c c  + a a b b c c  - a a a b b c c  -
       1 2 0 1 2 2 7     0 1 2 1 2 2 7    1 2 0 2 2 7    0 1 2 1 2 2 7  
     ------------------------------------------------------------------------
       2 4 2               4              2 4 2             2 3            
     2a a b b c c  + 2a a a b b b c c  + a a b b c c  + 2a a a b b b c c  -
       1 2 0 3 2 7     0 1 2 0 1 3 2 7    0 2 1 3 2 7     0 1 2 0 2 3 2 7  
     ------------------------------------------------------------------------
       2   3              2 2 2 2           2 4   2         3 3   2      
     4a a a b b b c c  + a a a b b c c  - 2a a b b c c  + 2a a b b c c  +
       0 1 2 1 2 3 2 7    0 1 2 2 3 2 7     0 2 0 3 2 7     0 2 2 3 2 7  
     ------------------------------------------------------------------------
       3 3 2             2 3               2   3 2           2   3          
     2a a b b c c  - 4a a a b b b c c  + 2a a a b b c c  + 2a a a b b b c c 
       1 2 0 4 2 7     0 1 2 0 1 4 2 7     0 1 2 1 4 2 7     0 1 2 0 3 4 2 7
     ------------------------------------------------------------------------
         3 3              2 2 2   2        3   2   2        4 2   2      
     - 2a a b b b c c  + a a a b b c c  - a a a b b c c  + a a b b c c  -
         0 2 1 3 4 2 7    0 1 2 0 4 2 7    0 1 2 1 4 2 7    0 2 3 4 2 7  
     ------------------------------------------------------------------------
       4 2 2             3 2               2 2 2 2             4            
     2a a b b c c  + 6a a a b b b c c  - 4a a a b b c c  - 2a a a b b b c c 
       1 2 0 5 2 7     0 1 2 0 1 5 2 7     0 1 2 1 5 2 7     0 1 2 0 2 5 2 7
     ------------------------------------------------------------------------
         2 3                 2 2 2               3   2            
     + 2a a a b b b c c  - 4a a a b b b c c  + 6a a a b b b c c  -
         0 1 2 1 2 5 2 7     0 1 2 0 3 5 2 7     0 1 2 1 3 5 2 7  
     ------------------------------------------------------------------------
       3 2                 4 2 2           2 3              
     2a a a b b b c c  - 2a a b b c c  - 2a a a b b b c c  +
       0 1 2 2 3 5 2 7     0 2 3 5 2 7     0 1 2 0 4 5 2 7  
     ------------------------------------------------------------------------
       3 2                 4                   2 4   2         3 3   2      
     2a a a b b b c c  - 2a a a b b b c c  + 2a a b b c c  - 2a a b b c c  +
       0 1 2 1 4 5 2 7     0 1 2 3 4 5 2 7     0 1 0 5 2 7     0 1 1 5 2 7  
     ------------------------------------------------------------------------
       4 2   2          5 2          2 4 2               4            
     2a a b b c c  + a a b b c c  - a a b b c c  - 2a a a b b b c c  +
       0 1 3 5 2 7    1 2 0 1 3 7    1 2 0 2 3 7     0 1 2 0 1 2 3 7  
     ------------------------------------------------------------------------
         2 3   2        2   3   2        2 2 2 3           5 2        
     2a a a b b c c  + a a a b b c c  - a a a b c c  - 2a a b b c c  +
       0 1 2 0 2 3 7    0 1 2 1 2 3 7    0 1 2 2 3 7     0 2 0 3 3 7  
     ------------------------------------------------------------------------
       2 4               3 3 2              4 2           2   3            
     4a a b b b c c  - 2a a b b c c  + a a a b b c c  - 2a a a b b b c c  +
       0 2 0 2 3 3 7     0 2 2 3 3 7    0 1 2 0 4 3 7     0 1 2 0 2 4 3 7  
     ------------------------------------------------------------------------
      3   2 2           2   3               2 2 2            
     a a a b b c c  + 2a a a b b b c c  - 2a a a b b b c c  -
      0 1 2 2 4 3 7     0 1 2 0 1 5 3 7     0 1 2 0 2 5 3 7  
     ------------------------------------------------------------------------
       3   2               3 2   2           3 3               4 2          
     2a a a b b b c c  + 2a a a b b c c  - 4a a b b b c c  + 4a a b b b c c 
       0 1 2 1 2 5 3 7     0 1 2 2 5 3 7     0 2 0 3 5 3 7     0 2 2 3 5 3 7
     ------------------------------------------------------------------------
         3   2               4                  4       2        4 2   2    
     + 2a a a b b b c c  - 2a a a b b b c c  + a a a b b c c  - a a b b c c 
         0 1 2 0 4 5 3 7     0 1 2 2 4 5 3 7    0 1 2 1 5 3 7    0 1 2 5 3 7
     ------------------------------------------------------------------------
         5     2        5     2        2 4 2              4   2      
     - 2a a b b c c  + a a b b c c  - a a b b c c  + a a a b b c c  +
         0 2 3 5 3 7    0 1 4 5 3 7    1 2 0 1 4 7    0 1 2 0 1 4 7  
     ------------------------------------------------------------------------
      3 3 2          2   3 2            3 2   2        2 2 2   2      
     a a b b c c  - a a a b b c c  - a a a b b c c  + a a a b b c c  -
      1 2 0 2 4 7    0 1 2 1 2 4 7    0 1 2 0 2 4 7    0 1 2 1 2 4 7  
     ------------------------------------------------------------------------
      2 4              2   3              3 3              3   2 2        
     a a b b b c c  + a a a b b b c c  + a a b b b c c  - a a a b b c c  +
      0 2 0 1 3 4 7    0 1 2 0 2 3 4 7    0 2 1 2 3 4 7    0 1 2 2 3 4 7  
     ------------------------------------------------------------------------
        2 3 2          2   3              2 2 2              3   2          
     a a a b b c c  - a a a b b b c c  - a a a b b b c c  + a a a b b b c c 
      0 1 2 0 4 4 7    0 1 2 0 1 4 4 7    0 1 2 0 2 4 4 7    0 1 2 1 2 4 4 7
     ------------------------------------------------------------------------
        3 3              4 2                 3 2 2          2 2 2            
     + a a b b b c c  - a a b b b c c  - 2a a a b b c c  + a a a b b b c c  +
        0 2 0 3 4 4 7    0 2 2 3 4 4 7     0 1 2 0 5 4 7    0 1 2 0 1 5 4 7  
     ------------------------------------------------------------------------
      3   2 2           2 3                 3 2              
     a a a b b c c  + 3a a a b b b c c  - 3a a a b b b c c  -
      0 1 2 1 5 4 7     0 1 2 0 2 5 4 7     0 1 2 1 2 5 4 7  
     ------------------------------------------------------------------------
       3   2              4 2               4                
     2a a a b b b c c  - a a b b b c c  + 3a a a b b b c c  +
       0 1 2 0 3 5 4 7    0 2 1 3 5 4 7     0 1 2 2 3 5 4 7  
     ------------------------------------------------------------------------
      3 2                4                  5                 3 3   2      
     a a a b b b c c  - a a a b b b c c  + a a b b b c c  - 2a a b b c c  +
      0 1 2 0 4 5 4 7    0 1 2 1 4 5 4 7    0 2 3 4 5 4 7     0 1 0 5 4 7  
     ------------------------------------------------------------------------
       4 2   2         5     2        3 3 2             2 3   2      
     2a a b b c c  - 2a a b b c c  + a a b b c c  - 2a a a b b c c  +
       0 1 1 5 4 7     0 1 3 5 4 7    1 2 0 1 5 7     0 1 2 0 1 5 7  
     ------------------------------------------------------------------------
      2   3 3        4 2 2             3 2              2 2 2 2        
     a a a b c c  - a a b b c c  + 2a a a b b b c c  - a a a b b c c  +
      0 1 2 1 5 7    1 2 0 2 5 7     0 1 2 0 1 2 5 7    0 1 2 1 2 5 7  
     ------------------------------------------------------------------------
         2 3 2          2   3              3 3 2           2 2 2            
     2a a a b b c c  - a a a b b b c c  - a a b b c c  - 3a a a b b b c c  +
       0 1 2 0 3 5 7    0 1 2 0 1 3 5 7    0 2 1 3 5 7     0 1 2 0 2 3 5 7  
     ------------------------------------------------------------------------
       3   2               3 3   2         4 2   2           3 2 2        
     3a a a b b b c c  + 2a a b b c c  - 2a a b b c c  - 3a a a b b c c  +
       0 1 2 1 2 3 5 7     0 2 0 3 5 7     0 2 2 3 5 7     0 1 2 0 4 5 7  
     ------------------------------------------------------------------------
       2 2 2               3   2 2          2 3              
     5a a a b b b c c  - 2a a a b b c c  + a a a b b b c c  -
       0 1 2 0 1 4 5 7     0 1 2 1 4 5 7    0 1 2 0 2 4 5 7  
     ------------------------------------------------------------------------
      3 2                 3   2               4 2            
     a a a b b b c c  - 3a a a b b b c c  + 2a a b b b c c  +
      0 1 2 1 2 4 5 7     0 1 2 0 3 4 5 7     0 2 1 3 4 5 7  
     ------------------------------------------------------------------------
      4                  3 2     2        4       2        5     2      
     a a a b b b c c  - a a a b b c c  + a a a b b c c  - a a b b c c  +
      0 1 2 2 3 4 5 7    0 1 2 0 4 5 7    0 1 2 1 4 5 7    0 2 3 4 5 7  
     ------------------------------------------------------------------------
         4   2           2 3                 3 2   2          2 4            
     4a a a b b c c  - 7a a a b b b c c  + 3a a a b b c c  - a a b b b c c  +
       0 1 2 0 5 5 7     0 1 2 0 1 5 5 7     0 1 2 1 5 5 7    0 1 0 2 5 5 7  
     ------------------------------------------------------------------------
      3 3               3 2                 4                  4 2          
     a a b b b c c  + 6a a a b b b c c  - 5a a a b b b c c  - a a b b b c c 
      0 1 1 2 5 5 7     0 1 2 0 3 5 5 7     0 1 2 1 3 5 5 7    0 1 2 3 5 5 7
     ------------------------------------------------------------------------
         5   2          3 3              4 2              5              
     + 2a a b b c c  + a a b b b c c  - a a b b b c c  + a a b b b c c  -
         0 2 3 5 5 7    0 1 0 4 5 5 7    0 1 1 4 5 5 7    0 1 3 4 5 5 7  
     ------------------------------------------------------------------------
         5 3          5 2               4 2           2 4            
     2a a b c c  + a a b b c c  + 5a a a b b c c  - 2a a b b b c c  -
       1 2 0 6 7    0 2 0 1 6 7     0 1 2 0 2 6 7     0 2 0 1 2 6 7  
     ------------------------------------------------------------------------
       2   3   2        3 3   2        3   2 3        2 4 2        
     4a a a b b c c  + a a b b c c  + a a a b c c  - a a b b c c  +
       0 1 2 0 2 6 7    0 2 1 2 6 7    0 1 2 2 6 7    0 2 0 4 6 7  
     ------------------------------------------------------------------------
       3 3              4 2 2           2   3 2           3 3            
     2a a b b b c c  - a a b b c c  - 4a a a b b c c  + 2a a b b b c c  +
       0 2 0 2 4 6 7    0 2 2 4 6 7     0 1 2 0 5 6 7     0 2 0 1 5 6 7  
     ------------------------------------------------------------------------
       3   2               4 2               4     2           4 2          
     6a a a b b b c c  - 2a a b b b c c  - 2a a a b b c c  - 2a a b b b c c 
       0 1 2 0 2 5 6 7     0 2 1 2 5 6 7     0 1 2 2 5 6 7     0 2 0 4 5 6 7
     ------------------------------------------------------------------------
         5                 4       2        5     2        5     2      
     + 2a a b b b c c  - 2a a a b b c c  + a a b b c c  + a a b b c c  -
         0 2 2 4 5 6 7     0 1 2 0 5 6 7    0 2 1 5 6 7    0 1 2 5 6 7  
     ------------------------------------------------------------------------
      6   2        2 4 3 2        4 2   2       2 3 2   2     2   3       2  
     a b b c c  + a a b c  - a a a b b c  - 2a a a b b c  + 2a a a b b b c  +
      0 4 5 6 7    1 2 0 7    0 1 2 0 1 7     0 1 2 0 2 7     0 1 2 0 1 2 7  
     ------------------------------------------------------------------------
      2 2 2   2 2    3   2   2 2    2 4 2   2     3 3       2    4 2 2   2  
     a a a b b c  - a a a b b c  + a a b b c  - 2a a b b b c  + a a b b c  +
      0 1 2 0 2 7    0 1 2 1 2 7    0 2 0 3 7     0 2 0 2 3 7    0 2 2 3 7  
     ------------------------------------------------------------------------
       2 2 2 2   2     3   2       2     3 2         2     4           2  
     2a a a b b c  - 2a a a b b b c  - 2a a a b b b c  + 2a a a b b b c  +
       0 1 2 0 5 7     0 1 2 0 1 5 7     0 1 2 0 2 5 7     0 1 2 1 2 5 7  
     ------------------------------------------------------------------------
       4 2       2     5         2    4 2   2 2    5     2 2    6   2 2  
     2a a b b b c  - 2a a b b b c  + a a b b c  - a a b b c  + a b b c  -
       0 2 0 3 5 7     0 2 2 3 5 7    0 1 0 5 7    0 1 1 5 7    0 3 5 7  
     ------------------------------------------------------------------------
      2 4 3         3 3 2           4 2   2        5   3      
     a a b c c  + 3a a b b c c  - 3a a b b c c  + a a b c c  +
      1 2 1 0 8     1 2 1 2 0 8     1 2 1 2 0 8    1 2 2 0 8  
     ------------------------------------------------------------------------
       2 4                   4 2           3 3                 2 3          
     3a a b b b c c  + 2a a a b b c c  - 3a a b b b c c  - 7a a a b b b c c 
       1 2 0 1 3 0 8     0 1 2 1 3 0 8     1 2 0 2 3 0 8     0 1 2 1 2 3 0 8
     ------------------------------------------------------------------------
           3 2 2               4   2        2 4   2         2   3   2      
     + 5a a a b b c c  - 4a a a b b c c  - a a b b c c  + 5a a a b b c c  -
         0 1 2 2 3 0 8     0 1 2 0 3 0 8    0 2 1 3 0 8     0 1 2 2 3 0 8  
     ------------------------------------------------------------------------
       3 3                2 3 2           4 2                3 2            
     3a a b b b c c  + a a a b b c c  + 3a a b b b c c  + a a a b b b c c  -
       1 2 0 1 4 0 8    0 1 2 1 4 0 8     1 2 0 2 4 0 8    0 1 2 1 2 4 0 8  
     ------------------------------------------------------------------------
         4   2             2 3               2   3            
     2a a a b b c c  + 5a a a b b b c c  - 2a a a b b b c c  -
       0 1 2 2 4 0 8     0 1 2 0 3 4 0 8     0 1 2 1 3 4 0 8  
     ------------------------------------------------------------------------
       2 2 2              3 3 2            3 2   2        2 3     2      
     3a a a b b b c c  + a a b b c c  - a a a b b c c  + a a a b b c c  +
       0 1 2 2 3 4 0 8    0 2 3 4 0 8    0 1 2 0 4 0 8    0 1 2 2 4 0 8  
     ------------------------------------------------------------------------
       4 2                 3 2 2           5                   4            
     3a a b b b c c  - 4a a a b b c c  - 3a a b b b c c  + 5a a a b b b c c 
       1 2 0 1 5 0 8     0 1 2 1 5 0 8     1 2 0 2 5 0 8     0 1 2 1 2 5 0 8
     ------------------------------------------------------------------------
          5 2             3 2               2 2 2            
     - a a b b c c  - 2a a a b b b c c  + 9a a a b b b c c  -
        0 1 2 5 0 8     0 1 2 0 3 5 0 8     0 1 2 1 3 5 0 8  
     ------------------------------------------------------------------------
       2 3                 3   2 2            4              
     7a a a b b b c c  - 6a a a b b c c  - a a a b b b c c  -
       0 1 2 2 3 5 0 8     0 1 2 3 5 0 8    0 1 2 0 4 5 0 8  
     ------------------------------------------------------------------------
      2 3                 2 4               3 2                3 3 2        
     a a a b b b c c  + 2a a b b b c c  + 3a a a b b b c c  - a a b b c c  +
      0 1 2 1 4 5 0 8     0 1 2 4 5 0 8     0 1 2 3 4 5 0 8    0 1 4 5 0 8  
     ------------------------------------------------------------------------
         5   2         2 4   2         3 3   2        2 4   2      
     2a a b b c c  - 2a a b b c c  + 2a a b b c c  + a a b b c c  -
       0 1 0 5 0 8     0 1 1 5 0 8     0 1 3 5 0 8    1 2 0 1 1 8  
     ------------------------------------------------------------------------
       3 3                2 3 2          4 2   2           3 2   2      
     2a a b b b c c  - a a a b b c c  + a a b b c c  + 2a a a b b c c  -
       1 2 0 1 2 1 8    0 1 2 1 2 1 8    1 2 0 2 1 8     0 1 2 1 2 1 8  
     ------------------------------------------------------------------------
        4   3         2 4 2               4                 2 3            
     a a a b c c  - 2a a b b c c  - 2a a a b b b c c  + 6a a a b b b c c  +
      0 1 2 2 1 8     1 2 0 3 1 8     0 1 2 0 1 3 1 8     0 1 2 0 2 3 1 8  
     ------------------------------------------------------------------------
       2   3               2 2 2 2           2 4   2         3 3   2      
     2a a a b b b c c  - 4a a a b b c c  + 2a a b b c c  - 2a a b b c c  +
       0 1 2 1 2 3 1 8     0 1 2 2 3 1 8     0 2 0 3 1 8     0 2 2 3 1 8  
     ------------------------------------------------------------------------
       3 3 2             3 2               2 3   2           2   3          
     2a a b b c c  - 4a a a b b b c c  + 2a a a b b c c  - 2a a a b b b c c 
       1 2 0 4 1 8     0 1 2 0 2 4 1 8     0 1 2 2 4 1 8     0 1 2 0 3 4 1 8
     ------------------------------------------------------------------------
         3   2              2 2 2   2        3 2     2         4 2 2        
     + 2a a a b b b c c  + a a a b b c c  - a a a b b c c  - 2a a b b c c  +
         0 1 2 2 3 4 1 8    0 1 2 0 4 1 8    0 1 2 2 4 1 8     1 2 0 5 1 8  
     ------------------------------------------------------------------------
         3 2              2 2 2 2             4              
     2a a a b b b c c  + a a a b b c c  + 2a a a b b b c c  -
       0 1 2 0 1 5 1 8    0 1 2 1 5 1 8     0 1 2 0 2 5 1 8  
     ------------------------------------------------------------------------
       2 3                2 4 2           2 2 2               3   2          
     4a a a b b b c c  + a a b b c c  - 4a a a b b b c c  - 2a a a b b b c c 
       0 1 2 1 2 5 1 8    0 1 2 5 1 8     0 1 2 0 3 5 1 8     0 1 2 1 3 5 1 8
     ------------------------------------------------------------------------
         3 2                 4 2 2           2 3              
     + 6a a a b b b c c  + 2a a b b c c  + 2a a a b b b c c  -
         0 1 2 2 3 5 1 8     0 2 3 5 1 8     0 1 2 0 4 5 1 8  
     ------------------------------------------------------------------------
       3 3               4                  4 2 2           2 4   2      
     2a a b b b c c  - 2a a a b b b c c  + a a b b c c  - 2a a b b c c  +
       0 1 2 4 5 1 8     0 1 2 3 4 5 1 8    0 1 4 5 1 8     0 1 0 5 1 8  
     ------------------------------------------------------------------------
       3 3   2         4 2   2        3 3   2          2 3 3      
     2a a b b c c  - 2a a b b c c  - a a b b c c  + a a a b c c  +
       0 1 1 5 1 8     0 1 3 5 1 8    1 2 0 1 2 8    0 1 2 1 2 8  
     ------------------------------------------------------------------------
       4 2                 3 2 2          5     2          4     2      
     2a a b b b c c  - 2a a a b b c c  - a a b b c c  + a a a b b c c  +
       1 2 0 1 2 2 8     0 1 2 1 2 2 8    1 2 0 2 2 8    0 1 2 1 2 2 8  
     ------------------------------------------------------------------------
       3 3 2            2 3               2   3 2             3 2            
     2a a b b c c  - a a a b b b c c  - 2a a a b b c c  - 3a a a b b b c c  +
       1 2 0 3 2 8    0 1 2 0 1 3 2 8     0 1 2 1 3 2 8     0 1 2 0 2 3 2 8  
     ------------------------------------------------------------------------
       2 2 2              2 3   2           2   3   2        3 3   2      
     5a a a b b b c c  - a a a b b c c  + 2a a a b b c c  + a a b b c c  -
       0 1 2 1 2 3 2 8    0 1 2 2 3 2 8     0 1 2 0 3 2 8    0 2 1 3 2 8  
     ------------------------------------------------------------------------
       3   2   2         4 2 2             3 2              2 2 2 2        
     3a a a b b c c  - 2a a b b c c  + 3a a a b b b c c  - a a a b b c c  +
       0 1 2 2 3 2 8     1 2 0 4 2 8     0 1 2 0 1 4 2 8    0 1 2 1 4 2 8  
     ------------------------------------------------------------------------
        4                2 3                 2 2 2            
     a a a b b b c c  - a a a b b b c c  - 3a a a b b b c c  +
      0 1 2 0 2 4 2 8    0 1 2 1 2 4 2 8     0 1 2 0 3 4 2 8  
     ------------------------------------------------------------------------
       3   2              3 2                4 2 2           5   2        
     2a a a b b b c c  + a a a b b b c c  - a a b b c c  + 2a a b b c c  -
       0 1 2 1 3 4 2 8    0 1 2 2 3 4 2 8    0 2 3 4 2 8     1 2 0 5 2 8  
     ------------------------------------------------------------------------
         4                 2 3   2            5              2 4            
     5a a a b b b c c  + 3a a a b b c c  + a a b b b c c  - a a b b b c c  +
       0 1 2 0 1 5 2 8     0 1 2 1 5 2 8    0 1 0 2 5 2 8    0 1 1 2 5 2 8  
     ------------------------------------------------------------------------
       2 3                 3 2                3 3               4     2      
     6a a a b b b c c  - 7a a a b b b c c  + a a b b b c c  + 4a a a b b c c 
       0 1 2 0 3 5 2 8     0 1 2 1 3 5 2 8    0 1 2 3 5 2 8     0 1 2 3 5 2 8
     ------------------------------------------------------------------------
        2 4              3 3              4 2              2 4 2        
     - a a b b b c c  + a a b b b c c  - a a b b b c c  - a a b b c c  +
        0 1 0 4 5 2 8    0 1 1 4 5 2 8    0 1 3 4 5 2 8    1 2 0 1 3 8  
     ------------------------------------------------------------------------
      3 3 2             2 3                 3 2   2        2 2 2   2      
     a a b b c c  + 2a a a b b b c c  - 2a a a b b c c  - a a a b b c c  +
      1 2 0 2 3 8     0 1 2 0 1 2 3 8     0 1 2 0 2 3 8    0 1 2 1 2 3 8  
     ------------------------------------------------------------------------
      2 3   3             4 2          2 4               2   3            
     a a a b c c  + 4a a a b b c c  - a a b b b c c  - 7a a a b b b c c  +
      0 1 2 2 3 8     0 1 2 0 3 3 8    0 2 0 1 3 3 8     0 1 2 0 2 3 3 8  
     ------------------------------------------------------------------------
      3 3               3   2 2             2 3 2          2   3            
     a a b b b c c  + 3a a a b b c c  - 3a a a b b c c  + a a a b b b c c  +
      0 2 1 2 3 3 8     0 1 2 2 3 3 8     0 1 2 0 4 3 8    0 1 2 0 1 4 3 8  
     ------------------------------------------------------------------------
       2 2 2              3   2               3 2   2          3 3          
     5a a a b b b c c  - a a a b b b c c  - 2a a a b b c c  + a a b b b c c 
       0 1 2 0 2 4 3 8    0 1 2 1 2 4 3 8     0 1 2 2 4 3 8    0 2 0 3 4 3 8
     ------------------------------------------------------------------------
        4 2              3   2   2        4       2           3 2 2        
     - a a b b b c c  - a a a b b c c  + a a a b b c c  + 2a a a b b c c  -
        0 2 2 3 4 3 8    0 1 2 0 4 3 8    0 1 2 2 4 3 8     0 1 2 0 5 3 8  
     ------------------------------------------------------------------------
       2 2 2              2 3                 3 2                3 3 2      
     3a a a b b b c c  - a a a b b b c c  + 3a a a b b b c c  - a a b b c c 
       0 1 2 0 1 5 3 8    0 1 2 0 2 5 3 8     0 1 2 1 2 5 3 8    0 1 2 5 3 8
     ------------------------------------------------------------------------
         3   2              4 2               4                
     + 6a a a b b b c c  - a a b b b c c  - 5a a a b b b c c  -
         0 1 2 0 3 5 3 8    0 2 1 3 5 3 8     0 1 2 2 3 5 3 8  
     ------------------------------------------------------------------------
       3 2                4                   4 2              5            
     3a a a b b b c c  + a a a b b b c c  + 2a a b b b c c  + a a b b b c c 
       0 1 2 0 4 5 3 8    0 1 2 1 4 5 3 8     0 1 2 4 5 3 8    0 2 3 4 5 3 8
     ------------------------------------------------------------------------
        5   2           3 3   2         4 2   2         5     2      
     - a a b b c c  + 2a a b b c c  - 2a a b b c c  + 2a a b b c c  +
        0 1 4 5 3 8     0 1 0 5 3 8     0 1 1 5 3 8     0 1 3 5 3 8  
     ------------------------------------------------------------------------
      3 3 2            2 3   2        4 2 2          2 2 2 2        
     a a b b c c  - a a a b b c c  - a a b b c c  + a a a b b c c  +
      1 2 0 1 4 8    0 1 2 0 1 4 8    1 2 0 2 4 8    0 1 2 1 2 4 8  
     ------------------------------------------------------------------------
        4     2        2 3     2           2 3 2           2   3            
     a a a b b c c  - a a a b b c c  - 2a a a b b c c  + 3a a a b b b c c  +
      0 1 2 0 2 4 8    0 1 2 1 2 4 8     0 1 2 0 3 4 8     0 1 2 0 1 3 4 8  
     ------------------------------------------------------------------------
      2 2 2               3   2              3 2   2           3 3   2      
     a a a b b b c c  - 3a a a b b b c c  + a a a b b c c  - 2a a b b c c  +
      0 1 2 0 2 3 4 8     0 1 2 1 2 3 4 8    0 1 2 2 3 4 8     0 2 0 3 4 8  
     ------------------------------------------------------------------------
       4 2   2          3 2 2          2 2 2              2 3              
     2a a b b c c  + a a a b b c c  - a a a b b b c c  - a a a b b b c c  +
       0 2 2 3 4 8    0 1 2 0 4 4 8    0 1 2 0 1 4 4 8    0 1 2 0 2 4 4 8  
     ------------------------------------------------------------------------
      3 2                3   2              4                
     a a a b b b c c  + a a a b b b c c  - a a a b b b c c  +
      0 1 2 1 2 4 4 8    0 1 2 0 3 4 4 8    0 1 2 2 3 4 4 8  
     ------------------------------------------------------------------------
      2 3                3 2   2          2 4              3 3            
     a a a b b b c c  - a a a b b c c  - a a b b b c c  + a a b b b c c  -
      0 1 2 0 1 5 4 8    0 1 2 1 5 4 8    0 1 0 2 5 4 8    0 1 1 2 5 4 8  
     ------------------------------------------------------------------------
       3 2                 4                  4 2               5   2        
     2a a a b b b c c  + 3a a a b b b c c  - a a b b b c c  - 2a a b b c c  +
       0 1 2 0 3 5 4 8     0 1 2 1 3 5 4 8    0 1 2 3 5 4 8     0 2 3 5 4 8  
     ------------------------------------------------------------------------
      3 3              4 2              5                4 2 2        
     a a b b b c c  - a a b b b c c  + a a b b b c c  - a a b b c c  +
      0 1 0 4 5 4 8    0 1 1 4 5 4 8    0 1 3 4 5 4 8    1 2 0 1 5 8  
     ------------------------------------------------------------------------
         3 2   2        2 2 2 3        5   2             4              
     2a a a b b c c  - a a a b c c  + a a b b c c  - 2a a a b b b c c  +
       0 1 2 0 1 5 8    0 1 2 1 5 8    1 2 0 2 5 8     0 1 2 0 1 2 5 8  
     ------------------------------------------------------------------------
      2 3   2           2 2 2               3   2 2           2 3            
     a a a b b c c  - 2a a a b b b c c  + 2a a a b b c c  + 2a a a b b b c c 
      0 1 2 1 2 5 8     0 1 2 0 1 3 5 8     0 1 2 1 3 5 8     0 1 2 0 2 3 5 8
     ------------------------------------------------------------------------
         3 2                4 2   2        4       2          4   2        
     - 2a a a b b b c c  - a a b b c c  + a a a b b c c  + a a a b b c c  -
         0 1 2 1 2 3 5 8    0 2 1 3 5 8    0 1 2 2 3 5 8    0 1 2 0 4 5 8  
     ------------------------------------------------------------------------
       2 3                3 2   2           3 2              
     2a a a b b b c c  + a a a b b c c  + 2a a a b b b c c  -
       0 1 2 0 1 4 5 8    0 1 2 1 4 5 8     0 1 2 0 3 4 5 8  
     ------------------------------------------------------------------------
       4                  5   2             5 2           2 4            
     2a a a b b b c c  + a a b b c c  - 2a a b b c c  + 4a a b b b c c  -
       0 1 2 1 3 4 5 8    0 2 3 4 5 8     0 1 0 5 5 8     0 1 0 1 5 5 8  
     ------------------------------------------------------------------------
       3 3 2           3 3               4 2               5   2        
     2a a b b c c  - 4a a b b b c c  + 4a a b b b c c  - 2a a b b c c  +
       0 1 1 5 5 8     0 1 0 3 5 5 8     0 1 1 3 5 5 8     0 1 3 5 5 8  
     ------------------------------------------------------------------------
       2 4 3             4 2          2 4   2           2 3 2        
     2a a b c c  - 2a a a b b c c  + a a b b c c  - 4a a a b b c c  +
       1 2 0 6 8     0 1 2 0 1 6 8    0 2 0 1 6 8     0 1 2 0 2 6 8  
     ------------------------------------------------------------------------
       2   3              3 3 2           2 2 2   2        3 2   3      
     2a a a b b b c c  - a a b b c c  + 3a a a b b c c  - a a a b c c  -
       0 1 2 0 1 2 6 8    0 2 1 2 6 8     0 1 2 0 2 6 8    0 1 2 2 6 8  
     ------------------------------------------------------------------------
       2 4 2           3 3               4 2 2           2   3 2        
     2a a b b c c  + 4a a b b b c c  - 2a a b b c c  + 4a a a b b c c  -
       0 2 0 3 6 8     0 2 0 2 3 6 8     0 2 2 3 6 8     0 1 2 0 4 6 8  
     ------------------------------------------------------------------------
       3 3               3   2               4 2               4     2      
     2a a b b b c c  - 6a a a b b b c c  + 2a a b b b c c  + 2a a a b b c c 
       0 2 0 1 4 6 8     0 1 2 0 2 4 6 8     0 2 1 2 4 6 8     0 1 2 2 4 6 8
     ------------------------------------------------------------------------
        4 2   2        5     2        4 2 2           4                
     + a a b b c c  - a a b b c c  + a a b b c c  - 2a a a b b b c c  +
        0 2 0 4 6 8    0 2 2 4 6 8    0 2 1 5 6 8     0 1 2 1 2 5 6 8  
     ------------------------------------------------------------------------
      4 2 2           4 2               5                 4                
     a a b b c c  - 4a a b b b c c  + 4a a b b b c c  + 4a a a b b b c c  -
      0 1 2 5 6 8     0 2 0 3 5 6 8     0 2 2 3 5 6 8     0 1 2 0 4 5 6 8  
     ------------------------------------------------------------------------
       5                 5                6 2           4 2   2      
     2a a b b b c c  - 2a a b b b c c  + a b b c c  - 2a a b b c c  +
       0 2 1 4 5 6 8     0 1 2 4 5 6 8    0 4 5 6 8     0 1 0 5 6 8  
     ------------------------------------------------------------------------
       5     2         6   2         3 3 3           2 3 2        
     2a a b b c c  - 2a b b c c  - 2a a b c c  + 3a a a b b c c  -
       0 1 1 5 6 8     0 3 5 6 8     1 2 0 7 8     0 1 2 0 1 7 8  
     ------------------------------------------------------------------------
      2   3   2           3 2 2           2 2 2              3   2 2        
     a a a b b c c  + 3a a a b b c c  - 4a a a b b b c c  + a a a b b c c  -
      0 1 2 0 1 7 8     0 1 2 0 2 7 8     0 1 2 0 1 2 7 8    0 1 2 1 2 7 8  
     ------------------------------------------------------------------------
      2 3     2        3 2     2         2   3 2          3 3            
     a a a b b c c  + a a a b b c c  - 2a a a b b c c  + a a b b b c c  +
      0 1 2 0 2 7 8    0 1 2 1 2 7 8     0 1 2 0 3 7 8    0 2 0 1 3 7 8  
     ------------------------------------------------------------------------
       3   2              4 2              4     2          2 2 2 2        
     3a a a b b b c c  - a a b b b c c  - a a a b b c c  - a a a b b c c  +
       0 1 2 0 2 3 7 8    0 2 1 2 3 7 8    0 1 2 2 3 7 8    0 1 2 0 4 7 8  
     ------------------------------------------------------------------------
      3   2              3 2                4                  4 2          
     a a a b b b c c  + a a a b b b c c  - a a a b b b c c  - a a b b b c c 
      0 1 2 0 1 4 7 8    0 1 2 0 2 4 7 8    0 1 2 1 2 4 7 8    0 2 0 3 4 7 8
     ------------------------------------------------------------------------
        5                 2 3   2           3 2                4     2      
     + a a b b b c c  - 2a a a b b c c  + 3a a a b b b c c  - a a a b b c c 
        0 2 2 3 4 7 8     0 1 2 0 5 7 8     0 1 2 0 1 5 7 8    0 1 2 1 5 7 8
     ------------------------------------------------------------------------
        3 3              4 2               4                  5              
     + a a b b b c c  - a a b b b c c  - 2a a a b b b c c  + a a b b b c c  +
        0 1 0 2 5 7 8    0 1 1 2 5 7 8     0 1 2 0 3 5 7 8    0 2 1 3 5 7 8  
     ------------------------------------------------------------------------
      5                4 2              5                6            
     a a b b b c c  - a a b b b c c  + a a b b b c c  - a b b b c c  +
      0 1 2 3 5 7 8    0 1 0 4 5 7 8    0 1 1 4 5 7 8    0 3 4 5 7 8  
     ------------------------------------------------------------------------
      4 2 3 2       3 2 2   2    2 2 2   2 2      4   2   2     2 3         2
     a a b c  - 2a a a b b c  + a a a b b c  - a a a b b c  + 2a a a b b b c 
      1 2 0 8     0 1 2 0 1 8    0 1 2 0 1 8    0 1 2 0 2 8     0 1 2 0 1 2 8
     ------------------------------------------------------------------------
        3 2   2   2     2 2 2 2   2     3   2       2     3 2         2  
     - a a a b b c  + 2a a a b b c  - 2a a a b b b c  - 2a a a b b b c  +
        0 1 2 1 2 8     0 1 2 0 3 8     0 1 2 0 1 3 8     0 1 2 0 2 3 8  
     ------------------------------------------------------------------------
       4           2    4 2   2 2    5     2 2    2 4 2   2     3 3       2  
     2a a a b b b c  + a a b b c  - a a b b c  + a a b b c  - 2a a b b b c  +
       0 1 2 1 2 3 8    0 2 0 3 8    0 2 2 3 8    0 1 0 5 8     0 1 0 1 5 8  
     ------------------------------------------------------------------------
      4 2 2   2     4 2       2     5         2    6 2   2    3 3 3      
     a a b b c  + 2a a b b b c  - 2a a b b b c  + a b b c  + a a b c c  -
      0 1 1 5 8     0 1 0 3 5 8     0 1 1 3 5 8    0 3 5 8    1 2 1 0 9  
     ------------------------------------------------------------------------
       4 2 2           5     2        6 3         3 3            
     3a a b b c c  + 3a a b b c c  - a b c c  - 3a a b b b c c  -
       1 2 1 2 0 9     1 2 1 2 0 9    1 2 0 9     1 2 0 1 3 0 9  
     ------------------------------------------------------------------------
         2 3 2           4 2                 3 2                 4   2      
     3a a a b b c c  + 3a a b b b c c  + 9a a a b b b c c  - 6a a a b b c c 
       0 1 2 1 3 0 9     1 2 0 2 3 0 9     0 1 2 1 2 3 0 9     0 1 2 2 3 0 9
     ------------------------------------------------------------------------
           2 3   2         2   3   2         2 2 2   2         3 3 3      
     + 6a a a b b c c  + 3a a a b b c c  - 9a a a b b c c  - 2a a b c c  +
         0 1 2 0 3 0 9     0 1 2 1 3 0 9     0 1 2 2 3 0 9     0 2 3 0 9  
     ------------------------------------------------------------------------
       4 2               5                   4                   5 2        
     3a a b b b c c  - 3a a b b b c c  - 3a a a b b b c c  + 3a a b b c c  -
       1 2 0 1 4 0 9     1 2 0 2 4 0 9     0 1 2 1 2 4 0 9     0 1 2 4 0 9  
     ------------------------------------------------------------------------
         3 2               2 3                 3   2 2        
     9a a a b b b c c  + 9a a a b b b c c  + 3a a a b b c c  +
       0 1 2 0 3 4 0 9     0 1 2 2 3 4 0 9     0 1 2 3 4 0 9  
     ------------------------------------------------------------------------
         4     2         2 4   2         3 2     2        3 3 3      
     3a a a b b c c  - 3a a b b c c  - 3a a a b b c c  + a a b c c  -
       0 1 2 0 4 0 9     0 1 2 4 0 9     0 1 2 3 4 0 9    0 1 4 0 9  
     ------------------------------------------------------------------------
       5                   4   2           6                 5            
     3a a b b b c c  + 3a a a b b c c  + 3a b b b c c  - 3a a b b b c c  +
       1 2 0 1 5 0 9     0 1 2 1 5 0 9     1 0 2 5 0 9     0 1 1 2 5 0 9  
     ------------------------------------------------------------------------
         4                 2 3                 2 4            
     6a a a b b b c c  - 9a a a b b b c c  + 3a a b b b c c  +
       0 1 2 0 3 5 0 9     0 1 2 1 3 5 0 9     0 1 2 3 5 0 9  
     ------------------------------------------------------------------------
       3 2   2             5               2 4               3 3            
     6a a a b b c c  - 3a a b b b c c  + 3a a b b b c c  - 3a a b b b c c  -
       0 1 2 3 5 0 9     0 1 0 4 5 0 9     0 1 1 4 5 0 9     0 1 3 4 5 0 9  
     ------------------------------------------------------------------------
      3 3   2         4 2                3 2 2          5     2      
     a a b b c c  + 2a a b b b c c  + a a a b b c c  - a a b b c c  -
      1 2 0 1 1 9     1 2 0 1 2 1 9    0 1 2 1 2 1 9    1 2 0 2 1 9  
     ------------------------------------------------------------------------
         4     2          5 3         3 3 2             2 3            
     2a a a b b c c  + a a b c c  + 2a a b b c c  + 3a a a b b b c c  -
       0 1 2 1 2 1 9    0 1 2 1 9     1 2 0 3 1 9     0 1 2 0 1 3 1 9  
     ------------------------------------------------------------------------
         3 2               2 2 2               2 3   2        
     7a a a b b b c c  - 3a a a b b b c c  + 5a a a b b c c  -
       0 1 2 0 2 3 1 9     0 1 2 1 2 3 1 9     0 1 2 2 3 1 9  
     ------------------------------------------------------------------------
       2   3   2        3 3   2         3   2   2         4 2 2        
     6a a a b b c c  + a a b b c c  + 5a a a b b c c  - 2a a b b c c  -
       0 1 2 0 3 1 9    0 2 1 3 1 9     0 1 2 2 3 1 9     1 2 0 4 1 9  
     ------------------------------------------------------------------------
        3 2                 4                2 3                 2 4 2      
     a a a b b b c c  + 5a a a b b b c c  + a a a b b b c c  - 3a a b b c c 
      0 1 2 0 1 4 1 9     0 1 2 0 2 4 1 9    0 1 2 1 2 4 1 9     0 1 2 4 1 9
     ------------------------------------------------------------------------
         2 2 2               3   2               3 2              
     + 9a a a b b b c c  - 2a a a b b b c c  - 7a a a b b b c c  -
         0 1 2 0 3 4 1 9     0 1 2 1 3 4 1 9     0 1 2 2 3 4 1 9  
     ------------------------------------------------------------------------
      4 2 2           2 3     2        3 2     2         3 3   2      
     a a b b c c  - 4a a a b b c c  + a a a b b c c  + 3a a b b c c  +
      0 2 3 4 1 9     0 1 2 0 4 1 9    0 1 2 1 4 1 9     0 1 2 4 1 9  
     ------------------------------------------------------------------------
       4       2        4 2 3         5   2            4              
     2a a a b b c c  - a a b c c  + 2a a b b c c  - a a a b b b c c  -
       0 1 2 3 4 1 9    0 1 4 1 9     1 2 0 5 1 9    0 1 2 0 1 5 1 9  
     ------------------------------------------------------------------------
      2 3   2             5               2 4               2 3              
     a a a b b c c  - 3a a b b b c c  + 3a a b b b c c  - 2a a a b b b c c  +
      0 1 2 1 5 1 9     0 1 0 2 5 1 9     0 1 1 2 5 1 9     0 1 2 0 3 5 1 9  
     ------------------------------------------------------------------------
       3 2                 3 3               4     2           2 4          
     5a a a b b b c c  - 3a a b b b c c  - 4a a a b b c c  + 3a a b b b c c 
       0 1 2 1 3 5 1 9     0 1 2 3 5 1 9     0 1 2 3 5 1 9     0 1 0 4 5 1 9
     ------------------------------------------------------------------------
         3 3               4 2              4 2   2          3 2 3      
     - 3a a b b b c c  + 3a a b b b c c  + a a b b c c  - a a a b c c  -
         0 1 1 4 5 1 9     0 1 3 4 5 1 9    1 2 0 1 2 9    0 1 2 1 2 9  
     ------------------------------------------------------------------------
       5                   4   2          6   2          5   2      
     2a a b b b c c  + 2a a a b b c c  + a b b c c  - a a b b c c  -
       1 2 0 1 2 2 9     0 1 2 1 2 2 9    1 0 2 2 9    0 1 1 2 2 9  
     ------------------------------------------------------------------------
       4 2 2           2 2 2 2             4                 2 3            
     2a a b b c c  + 3a a a b b c c  + 4a a a b b b c c  - 6a a a b b b c c 
       1 2 0 3 2 9     0 1 2 1 3 2 9     0 1 2 0 2 3 2 9     0 1 2 1 2 3 2 9
     ------------------------------------------------------------------------
        2 4 2           3   2   2         3 2     2         4 2 3      
     + a a b b c c  - 4a a a b b c c  + 4a a a b b c c  + 2a a b c c  +
        0 1 2 3 2 9     0 1 2 1 3 2 9     0 1 2 2 3 2 9     0 2 3 2 9  
     ------------------------------------------------------------------------
       5   2             4                   5               2 4            
     2a a b b c c  - 2a a a b b b c c  - 2a a b b b c c  + 2a a b b b c c  +
       1 2 0 4 2 9     0 1 2 0 1 4 2 9     0 1 0 2 4 2 9     0 1 1 2 4 2 9  
     ------------------------------------------------------------------------
       3 2                 3 3               4     2          2 4   2      
     2a a a b b b c c  - 2a a b b b c c  - 2a a a b b c c  + a a b b c c  -
       0 1 2 1 3 4 2 9     0 1 2 3 4 2 9     0 1 2 3 4 2 9    0 1 0 4 2 9  
     ------------------------------------------------------------------------
      3 3   2        4 2   2         6 2             5            
     a a b b c c  + a a b b c c  - 2a b b c c  + 4a a b b b c c  -
      0 1 1 4 2 9    0 1 3 4 2 9     1 0 5 2 9     0 1 0 1 5 2 9  
     ------------------------------------------------------------------------
       2 4 2           2 4               3 3               4 2 2        
     2a a b b c c  - 4a a b b b c c  + 4a a b b b c c  - 2a a b b c c  +
       0 1 1 5 2 9     0 1 0 3 5 2 9     0 1 1 3 5 2 9     0 1 3 5 2 9  
     ------------------------------------------------------------------------
      3 3 2          4 2 2             3 2                 4     2      
     a a b b c c  - a a b b c c  - 2a a a b b b c c  + 2a a a b b c c  +
      1 2 0 1 3 9    1 2 0 2 3 9     0 1 2 0 1 2 3 9     0 1 2 0 2 3 9  
     ------------------------------------------------------------------------
      2 3     2        2 4 3           2 3 2           2   3            
     a a a b b c c  - a a b c c  - 6a a a b b c c  + 3a a a b b b c c  -
      0 1 2 1 2 3 9    0 1 2 3 9     0 1 2 0 3 3 9     0 1 2 0 1 3 3 9  
     ------------------------------------------------------------------------
      3 3 2           2 2 2              3   2               3 2   2        
     a a b b c c  + 9a a a b b b c c  - a a a b b b c c  - 4a a a b b c c  +
      0 2 1 3 3 9     0 1 2 0 2 3 3 9    0 1 2 1 2 3 3 9     0 1 2 2 3 3 9  
     ------------------------------------------------------------------------
       3 3   2         4 2   2           3 2 2           2 2 2            
     2a a b b c c  - 2a a b b c c  + 5a a a b b c c  - 3a a a b b b c c  +
       0 2 0 3 3 9     0 2 2 3 3 9     0 1 2 0 4 3 9     0 1 2 0 1 4 3 9  
     ------------------------------------------------------------------------
      3   2 2           2 3                3 2                 3 3 2        
     a a a b b c c  - 7a a a b b b c c  + a a a b b b c c  + 3a a b b c c  -
      0 1 2 1 4 3 9     0 1 2 0 2 4 3 9    0 1 2 1 2 4 3 9     0 1 2 4 3 9  
     ------------------------------------------------------------------------
       3   2               4 2               4                
     7a a a b b b c c  + 2a a b b b c c  + 5a a a b b b c c  +
       0 1 2 0 3 4 3 9     0 2 1 3 4 3 9     0 1 2 2 3 4 3 9  
     ------------------------------------------------------------------------
       3 2     2         4       2         4 2   2        5     2      
     5a a a b b c c  - 2a a a b b c c  - 3a a b b c c  - a a b b c c  +
       0 1 2 0 4 3 9     0 1 2 1 4 3 9     0 1 2 4 3 9    0 2 3 4 3 9  
     ------------------------------------------------------------------------
      5   3           4   2           2 3                3 2   2        
     a a b c c  - 4a a a b b c c  + 5a a a b b b c c  - a a a b b c c  +
      0 1 4 3 9     0 1 2 0 5 3 9     0 1 2 0 1 5 3 9    0 1 2 1 5 3 9  
     ------------------------------------------------------------------------
       2 4               3 3               3 2                4              
     3a a b b b c c  - 3a a b b b c c  - 2a a a b b b c c  - a a a b b b c c 
       0 1 0 2 5 3 9     0 1 1 2 5 3 9     0 1 2 0 3 5 3 9    0 1 2 1 3 5 3 9
     ------------------------------------------------------------------------
         4 2               5   2           3 3               4 2            
     + 3a a b b b c c  + 2a a b b c c  - 3a a b b b c c  + 3a a b b b c c  -
         0 1 2 3 5 3 9     0 2 3 5 3 9     0 1 0 4 5 3 9     0 1 1 4 5 3 9  
     ------------------------------------------------------------------------
       5                4 2 2            3 2   2        5   2        
     3a a b b b c c  - a a b b c c  + a a a b b c c  + a a b b c c  -
       0 1 3 4 5 3 9    1 2 0 1 4 9    0 1 2 0 1 4 9    1 2 0 2 4 9  
     ------------------------------------------------------------------------
      2 3   2            5   2        2 4   2           3 2 2        
     a a a b b c c  - a a b b c c  + a a b b c c  + 4a a a b b c c  -
      0 1 2 1 2 4 9    0 1 0 2 4 9    0 1 1 2 4 9     0 1 2 0 3 4 9  
     ------------------------------------------------------------------------
       2 2 2              3   2 2           2 3              
     6a a a b b b c c  + a a a b b c c  - 2a a a b b b c c  +
       0 1 2 0 1 3 4 9    0 1 2 1 3 4 9     0 1 2 0 2 3 4 9  
     ------------------------------------------------------------------------
       3 2                3 3 2           3   2   2        4 2   2      
     4a a a b b b c c  - a a b b c c  + 4a a a b b c c  - a a b b c c  -
       0 1 2 1 2 3 4 9    0 1 2 3 4 9     0 1 2 0 3 4 9    0 2 1 3 4 9  
     ------------------------------------------------------------------------
       4       2           4   2           2 3                3 2   2        
     3a a a b b c c  - 3a a a b b c c  + 4a a a b b b c c  - a a a b b c c  +
       0 1 2 2 3 4 9     0 1 2 0 4 4 9     0 1 2 0 1 4 4 9    0 1 2 1 4 4 9  
     ------------------------------------------------------------------------
       2 4               3 3               3 2                 4 2          
     2a a b b b c c  - 2a a b b b c c  - 2a a a b b b c c  + 2a a b b b c c 
       0 1 0 2 4 4 9     0 1 1 2 4 4 9     0 1 2 0 3 4 4 9     0 1 2 3 4 4 9
     ------------------------------------------------------------------------
        5   2          3 3   2        4 2   2        5     2      
     + a a b b c c  - a a b b c c  + a a b b c c  - a a b b c c  +
        0 2 3 4 4 9    0 1 0 4 4 9    0 1 1 4 4 9    0 1 3 4 4 9  
     ------------------------------------------------------------------------
         5 2           2 4               3 3 2           3 3            
     2a a b b c c  - 4a a b b b c c  + 2a a b b c c  + 4a a b b b c c  -
       0 1 0 5 4 9     0 1 0 1 5 4 9     0 1 1 5 4 9     0 1 0 3 5 4 9  
     ------------------------------------------------------------------------
       4 2               5   2          5   2             4     2      
     4a a b b b c c  + 2a a b b c c  + a a b b c c  - 2a a a b b c c  +
       0 1 1 3 5 4 9     0 1 3 5 4 9    1 2 0 1 5 9     0 1 2 0 1 5 9  
     ------------------------------------------------------------------------
      2 3   3        6 2             5              2 4 2        
     a a a b c c  - a b b c c  + 2a a b b b c c  - a a b b c c  -
      0 1 2 1 5 9    1 0 2 5 9     0 1 0 1 2 5 9    0 1 1 2 5 9  
     ------------------------------------------------------------------------
         4   2           2 3                 3 2   2           2 4          
     2a a a b b c c  + 6a a a b b b c c  - 4a a a b b c c  - 2a a b b b c c 
       0 1 2 0 3 5 9     0 1 2 0 1 3 5 9     0 1 2 1 3 5 9     0 1 0 2 3 5 9
     ------------------------------------------------------------------------
         3 3               3 2     2         4       2        4 2   2      
     + 2a a b b b c c  - 4a a a b b c c  + 5a a a b b c c  - a a b b c c  -
         0 1 1 2 3 5 9     0 1 2 0 3 5 9     0 1 2 1 3 5 9    0 1 2 3 5 9  
     ------------------------------------------------------------------------
       5   3          5 2           2 4              3 3 2        
     2a a b c c  + a a b b c c  - 2a a b b b c c  + a a b b c c  +
       0 2 3 5 9    0 1 0 4 5 9     0 1 0 1 4 5 9    0 1 1 4 5 9  
     ------------------------------------------------------------------------
       3 3               4 2              5   2           3 3 3      
     2a a b b b c c  - 2a a b b b c c  + a a b b c c  - 2a a b c c  +
       0 1 0 3 4 5 9     0 1 1 3 4 5 9    0 1 3 4 5 9     1 2 0 6 9  
     ------------------------------------------------------------------------
         2 3 2           2   3   2        3 3 3           3 2 2        
     3a a a b b c c  - 3a a a b b c c  + a a b c c  + 3a a a b b c c  -
       0 1 2 0 1 6 9     0 1 2 0 1 6 9    0 2 1 6 9     0 1 2 0 2 6 9  
     ------------------------------------------------------------------------
       2 3     2        3 3 3         2   3 2           3 3            
     3a a a b b c c  + a a b c c  + 6a a a b b c c  - 3a a b b b c c  -
       0 1 2 0 2 6 9    0 1 2 6 9     0 1 2 0 3 6 9     0 2 0 1 3 6 9  
     ------------------------------------------------------------------------
       3   2               4 2               4     2           2 2 2 2      
     9a a a b b b c c  + 3a a b b b c c  + 3a a a b b c c  - 9a a a b b c c 
       0 1 2 0 2 3 6 9     0 2 1 2 3 6 9     0 1 2 2 3 6 9     0 1 2 0 4 6 9
     ------------------------------------------------------------------------
         3   2               4 2 2           3 2              
     + 9a a a b b b c c  - 3a a b b c c  + 9a a a b b b c c  -
         0 1 2 0 1 4 6 9     0 2 1 4 6 9     0 1 2 0 2 4 6 9  
     ------------------------------------------------------------------------
       4                   4 2 2           4 2               5              
     3a a a b b b c c  - 3a a b b c c  + 3a a b b b c c  - 3a a b b b c c  -
       0 1 2 1 2 4 6 9     0 1 2 4 6 9     0 2 0 3 4 6 9     0 2 2 3 4 6 9  
     ------------------------------------------------------------------------
       4       2         5     2         5     2        6 3      
     6a a a b b c c  + 3a a b b c c  + 3a a b b c c  - a b c c  +
       0 1 2 0 4 6 9     0 2 1 4 6 9     0 1 2 4 6 9    0 4 6 9  
     ------------------------------------------------------------------------
       2 3   2           3 2                 4     2           3 3          
     6a a a b b c c  - 9a a a b b b c c  + 3a a a b b c c  - 3a a b b b c c 
       0 1 2 0 5 6 9     0 1 2 0 1 5 6 9     0 1 2 1 5 6 9     0 1 0 2 5 6 9
     ------------------------------------------------------------------------
         4 2               4                   5              
     + 3a a b b b c c  + 6a a a b b b c c  - 3a a b b b c c  -
         0 1 1 2 5 6 9     0 1 2 0 3 5 6 9     0 2 1 3 5 6 9  
     ------------------------------------------------------------------------
       5                 4 2               5                 6            
     3a a b b b c c  + 3a a b b b c c  - 3a a b b b c c  + 3a b b b c c  +
       0 1 2 3 5 6 9     0 1 0 4 5 6 9     0 1 1 4 5 6 9     0 3 4 5 6 9  
     ------------------------------------------------------------------------
       4 2 3           3 2 2           2 2 2   2        3   2 3      
     2a a b c c  - 4a a a b b c c  + 3a a a b b c c  - a a a b c c  -
       1 2 0 7 9     0 1 2 0 1 7 9     0 1 2 0 1 7 9    0 1 2 1 7 9  
     ------------------------------------------------------------------------
         4   2           2 3                2 4   2        3 3   2      
     2a a a b b c c  + 2a a a b b b c c  + a a b b c c  - a a b b c c  +
       0 1 2 0 2 7 9     0 1 2 0 1 2 7 9    0 1 0 2 7 9    0 1 1 2 7 9  
     ------------------------------------------------------------------------
      4 2 2           4                  4 2 2           4 2   2      
     a a b b c c  - 2a a a b b b c c  + a a b b c c  - 2a a b b c c  +
      0 2 1 3 7 9     0 1 2 1 2 3 7 9    0 1 2 3 7 9     0 2 0 3 7 9  
     ------------------------------------------------------------------------
       5     2         2 3   2           3 2                 4     2        
     2a a b b c c  + 4a a a b b c c  - 6a a a b b b c c  + 2a a a b b c c  -
       0 2 2 3 7 9     0 1 2 0 4 7 9     0 1 2 0 1 4 7 9     0 1 2 1 4 7 9  
     ------------------------------------------------------------------------
       3 3               4 2               4                   5            
     2a a b b b c c  + 2a a b b b c c  + 4a a a b b b c c  - 2a a b b b c c 
       0 1 0 2 4 7 9     0 1 1 2 4 7 9     0 1 2 0 3 4 7 9     0 2 1 3 4 7 9
     ------------------------------------------------------------------------
         5                4 2   2        5     2        6   2      
     - 2a a b b b c c  + a a b b c c  - a a b b c c  + a b b c c  -
         0 1 2 3 4 7 9    0 1 0 4 7 9    0 1 1 4 7 9    0 3 4 7 9  
     ------------------------------------------------------------------------
       2 4 2           3 3               4 2 2           4 2            
     2a a b b c c  + 4a a b b b c c  - 2a a b b c c  - 4a a b b b c c  +
       0 1 0 5 7 9     0 1 0 1 5 7 9     0 1 1 5 7 9     0 1 0 3 5 7 9  
     ------------------------------------------------------------------------
       5                 6 2           5   3           4   2        
     4a a b b b c c  - 2a b b c c  - 2a a b c c  + 5a a a b b c c  -
       0 1 1 3 5 7 9     0 3 5 7 9     1 2 0 8 9     0 1 2 0 1 8 9  
     ------------------------------------------------------------------------
       2 3     2        3 2   3          5 2           2 4            
     4a a a b b c c  + a a a b c c  + a a b b c c  - 2a a b b b c c  +
       0 1 2 0 1 8 9    0 1 2 1 8 9    0 1 0 2 8 9     0 1 0 1 2 8 9  
     ------------------------------------------------------------------------
      3 3 2           2 3   2           3 2                 4     2        
     a a b b c c  - 4a a a b b c c  + 6a a a b b b c c  - 2a a a b b c c  +
      0 1 1 2 8 9     0 1 2 0 3 8 9     0 1 2 0 1 3 8 9     0 1 2 1 3 8 9  
     ------------------------------------------------------------------------
       3 3               4 2               4       2        5     2      
     2a a b b b c c  - 2a a b b b c c  - 2a a a b b c c  + a a b b c c  +
       0 1 0 2 3 8 9     0 1 1 2 3 8 9     0 1 2 0 3 8 9    0 2 1 3 8 9  
     ------------------------------------------------------------------------
      5     2        2 4 2           3 3              4 2 2        
     a a b b c c  - a a b b c c  + 2a a b b b c c  - a a b b c c  -
      0 1 2 3 8 9    0 1 0 4 8 9     0 1 0 1 4 8 9    0 1 1 4 8 9  
     ------------------------------------------------------------------------
       4 2               5                6 2          6 3 2       5 2   2  
     2a a b b b c c  + 2a a b b b c c  - a b b c c  + a b c  - 3a a b b c  +
       0 1 0 3 4 8 9     0 1 1 3 4 8 9    0 3 4 8 9    1 0 9     0 1 0 1 9  
     ------------------------------------------------------------------------
       2 4   2 2    3 3 3 2     2 4 2   2     3 3       2     4 2 2   2  
     3a a b b c  - a a b c  + 3a a b b c  - 6a a b b b c  + 3a a b b c  +
       0 1 0 1 9    0 1 1 9     0 1 0 3 9     0 1 0 1 3 9     0 1 1 3 9  
     ------------------------------------------------------------------------
       4 2   2 2     5     2 2    6 3 2
     3a a b b c  - 3a a b b c  + a b c
       0 1 0 3 9     0 1 1 3 9    0 3 9

o5 : ZZ[a , a , a , b , b , b , b , b , b , c , c , c , c , c , c , c , c , c , c ]
         0   1   2   0   1   2   3   4   5   0   1   2   3   4   5   6   7   8   9

On page 88 of Using Algebraic Geometry is stated that the resultant of three ternary quadrics, written out in its full glory, has 21.894 terms.

i6 : F = genericPolynomials({2,2,2},ZZ)

         2               2                        2     2               2  
o6 = {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     2               2                        2
     b x x  + b x x  + b x , c x  + c x x  + c x  + c x x  + c x x  + c x }
      2 0 2    4 1 2    5 2   0 0    1 0 1    3 1    2 0 2    4 1 2    5 2

o6 : List
i7 : time # terms resultant F
     -- used 0.421821 seconds

o7 = 21894

See also

Ways to use resultant :