This notebook calculates the polynomial S in the case 
d_  
        1  =  d_  2  =  d_  3  = 
          5.   It is recommended that the result is 
stored and the checks on the coefficients performed separately.

(x*H1  +  y*H2  +  z*H3)^3  (a1*H1  + b1*H2  + 
            c1*H3) (a2*H1  + b2*H2  + c2*H3) (a3*H1  + b3*H2  + 
            c3*H3) (a4*H1  + b4*H2  + c4*H3) (a5*H1  + b5*H2  + 
            c5*H3) (a6*H1  + b6*H2  + c6*H3) (a7*H1  +  
            b7*H2  +  c7*H3)  (a8*H1  +  b8*H2  +  
            c8*H3) (a9*H1  +  b9*H2  +  c9*H3) (a10*H1  +  
            b10*H2  +  c10*H3) (a11*H1  +  b11*H2  +  
            c11*H3) (a12*H1  +  b12*H2  +  c12*H3);

P  =  Expand[%];

F  =  
        P /. {H1^15  -> 0, H1^14  -> 0, H1^13  -> 0,  
            H1^12  -> 0,  H1^11  -> 0, H1^10  -> 0,  
            H1^9  -> 0,  H1^8  -> 0,  H1^7  -> 0,  
            H1^6  -> 0,   H2^15  -> 0 , H2^14  -> 0 , 
            H2^13  -> 0 , H2^12  -> 0 ,  H2^11  -> 0, 
            H2^10  -> 0,  H2^9  -> 0,  H2^8  -> 0,  
            H2^7  -> 0,  H2^6  ->  0,  H3^15  -> 0,  
            H3^14  -> 0,  H3^13  -> 0, H3^11  -> 0,  
            H3^10  -> 0,  H3^9  -> 0,  H3^8  -> 0,  
            H3^7  -> 0, H3^6 -> 0,   H1^5  ->  1,  
            H2^5  ->  1,  H3 ^5 -> 1};

a300  =  
      F /. {x -> 1,  y -> 0,  z -> 0 };a030  =  
      F /. {x -> 0,  y -> 1,  z -> 0 };a003  =  
      F /. {x -> 0,  y -> 0,  z -> 1 };G  =  
      F  -  a300*x^3  -  a030*y^3  - a003*z^3;Q =  
      G /. {z -> 0};Q /. {y^2  -> 0,  x -> 1,  
        y  -> 1};

a210  =  %/3; 
    Q /. {x^2  -> 0,  x -> 1,  y  -> 1};

a120  =  %/3; R  =  G  -  Q; S  =  R /. {y -> 0}; 
    S /. {z^2  -> 0,  x -> 1,  z -> 1};

a201  =  %/3; 
    S /. {x^2  -> 0,  x -> 1,  z -> 1};

a102  =  %/3 ;T  =  R  - S ;T0  =  
        T /. {x -> 0} ;T0 /. {z^2  -> 0,  
          y -> 1,  z -> 1} ;

a021  =  %/3 ;T0 /. {y^2  -> 0,  
          y -> 1,  z -> 1} ;

a012  =  %/3; (T  -  T0) /. {x -> 1 ,  y -> 1,  
        z -> 1};

a111  =  %/6; Q = 0; R = 0; T = 0; T0 = 0; G = 0; P = 0;

S =  
        a300 a120 a021 a003   -  a300 a120 a012^2  -  
          a300 a111 a030 a003  +  a300 a111 a021 a012  +  
          a300 a102 a030 a012  -  a300 a102 a021^2  -  
          a210^2 a021 a003  +  a210^2 a012^2  +  
          a210 a201 a030 a003  -  a210 a201 a021 a012  +  
          a210 a120 a111 a003  -  a210 a120 a102 a012   -  
          2 a210 a111^2 a012  +  3 a210 a111 a102 a021  -  
          a210 a102^2 a030  -  a201^2 a030 a012   +  
          a201^2 a021^2  -  a201 a120^2 a003  +  
          3 a201 a120 a111 a012  -  a201 a120 a102 a021  -  
          2 a201 a111^2 a021  +  a201 a111 a102 a030  +  
          a120^2 a102^2   -  2 a120 a111^2 a102  +  
          a111^4;

T = S;

S  >>  tsave