Die Rechnung in Singular

ring r=0,(a(0..7),e(1..3),q(1..3),n,v),dp;
ideal i=
    n*v*q(1)*q(2)*q(3)            +n*(1-v)*e(1)*e(2)*e(3)            -a(7),
    n*v*q(1)*q(2)*(1-q(3))        +n*(1-v)*e(1)*e(2)*(1-e(3))        -a(6),    
    n*v*q(1)*q(3)*(1-q(2))        +n*(1-v)*e(1)*e(3)*(1-e(2))        -a(5),
    n*v*q(1)*(1-q(2))*(1-q(3))    +n*(1-v)*e(1)*(1-e(2))*(1-e(3))    -a(4),
    n*v*q(3)*q(2)*(1-q(1))        +n*(1-v)*e(3)*e(2)*(1-e(1))        -a(3),
    n*v*q(2)*(1-q(1))*(1-q(3))    +n*(1-v)*e(2)*(1-e(1))*(1-e(3))    -a(2),
    n*v*q(3)*(1-q(1))*(1-q(2))    +n*(1-v)*e(3)*(1-e(1))*(1-e(2))    -a(1),
    n*v*(1-q(1))*(1-q(2))*(1-q(3))+n*(1-v)*(1-e(1))*(1-e(2))*(1-e(3))-a(0);

ideal k=eliminate(i,n*q(1)*q(2)*q(3)*e(1)*e(2)*e(3));

matrix m=coeffs(k[1],v); 
m[2,1]+m[3,1];
//->0
   
list fac1=factorize(m[3,1]);          //f1^2*f4
fac1;
[1]:
   _[1]=1
   _[2]=a(0)+a(1)+a(2)+a(3)+a(4)+a(5)+a(6)+a(7)   
   _[3]=a(3)^2*a(4)^2-2*a(2)*a(3)*a(4)*a(5)+a(2)^2*a(5)^2-2*a(1)*a(3)*a(4)*a(6)
        -2*a(1)*a(2)*a(5)*a(6)+4*a(0)*a(3)*a(5)*a(6)+a(1)^2*a(6)^2
        +4*a(1)*a(2)*a(4)*a(7)-2*a(0)*a(3)*a(4)*a(7)-2*a(0)*a(2)*a(5)*a(7)
        -2*a(0)*a(1)*a(6)*a(7)+a(0)^2*a(7)^2 
[2]:
   1,2,1

list fac2=factorize(m[3,1]-4*m[1,1]);   //f3^2
fac2;
[1]:
   _[1]=1
   _[2]=-2*a(1)*a(2)*a(4)+a(0)*a(3)*a(4)-a(1)*a(3)*a(4)-a(2)*a(3)*a(4)
        -a(3)^2*a(4)+a(3)*a(4)^2+a(0)*a(2)*a(5)-a(1)*a(2)*a(5)+a(2)^2*a(5)
        +2*a(0)*a(3)*a(5)+a(2)*a(3)*a(5)-a(2)*a(4)*a(5)+a(3)*a(4)*a(5)-a(2)*a(5)^2
        +a(0)*a(1)*a(6)+a(1)^2*a(6)-a(1)*a(2)*a(6)+2*a(0)*a(3)*a(6)+a(1)*a(3)*a(6)
        -a(1)*a(4)*a(6)+a(3)*a(4)*a(6)+2*a(0)*a(5)*a(6)+a(1)*a(5)*a(6)+a(2)*a(5)*a(6)
        +2*a(3)*a(5)*a(6)-a(1)*a(6)^2-a(0)^2*a(7)-a(0)*a(1)*a(7)-a(0)*a(2)*a(7)
        -2*a(1)*a(2)*a(7)+a(0)*a(3)*a(7)-a(0)*a(4)*a(7)-2*a(1)*a(4)*a(7)
        -2*a(2)*a(4)*a(7)-a(3)*a(4)*a(7)+a(0)*a(5)*a(7)-a(2)*a(5)*a(7)+a(0)*a(6)*a(7)
        -a(1)*a(6)*a(7)+a(0)*a(7)^2
[2]:
   1,2