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Computation 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];
list fac1=factorize(m[3,1]); //f1^2*f4
fac1;
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==>
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[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
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list fac2=factorize(m[3,1]-4*m[1,1]); //f3^2
fac2;
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==>
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[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
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