I have the following variables
p = (Γ + κ1 + κ2)/2;
u1 = 36 g1^2 (-2 p + 3 κ2) + (36 g2^2 + (2 p - 3 κ1) (2 p - 3 κ2)) (4 p - 3 (κ1 + κ2));
u2 = 2^(2/3)*(12 g1^2 + 12 g2^2 - 4 p^2 + 6 p (κ1 + κ2) - 3 (κ1^2 + κ1 κ2 + κ2^2));
sig = ((u1 + Sqrt[u1^2 + u2^3])/16)^(1/3);
and I define my functions as
o1 = 1/3*(p + sig - u2/(2^(8/3)*sig) - 3*I*ω);
o2 = 1/3*(p - 1/2*(sig - u2/(2^(8/3)*sig)) + (I*Sqrt[3])/2*(sig + u2/(2^(8/3)*sig)) - 3*I*ω);
o3 = 1/3*(p - 1/2*(sig - u2/(2^(8/3)*sig)) - (I*Sqrt[3])/2*(sig + u2/(2^(8/3)*sig)) - 3*I*ω);
Now, I wish to find the common value of g1 for which 'o1 = o2 = o3'. I do the following
Solve[o1==o2==o3,g1]//Simplify
And I'm returned with two values of such g1
{{g1 -> -(Sqrt[-12 g2^2 + Γ^2 + κ1^2 - κ1κ2 + κ2^2 - Γ (κ1 + κ2)]/(2 Sqrt[3]))},
{g1 -> Sqrt[-12 g2^2 + Γ^2 + κ1^2 - κ1κ2 + κ2^2 - Γ (κ1 + κ2)]/(2 Sqrt[3])}}
But upon checking by substituting them back into o1, o2 and o3 (for the first common g1), they do not yield the same result
tei1 = (o1 /. {g1 -> -(Sqrt[-12 g2^2 + Γ^2 + κ1^2 - κ1κ2 + κ2^2 - Γ (κ1 + κ2)]/(2 Sqrt[3]))}) // FullSimplify
tei2 = o2 /. {g1 -> -(Sqrt[-12 g2^2 + Γ^2 + κ1^2 - κ1κ2 + κ2^2 - Γ (κ1 + κ2)]/(2 Sqrt[3]))} // FullSimplify
Doing
(tei1-tei2)//FullSimplify
Gives
((3 - I Sqrt[3]) (-(Γ + κ1 - 2 κ2)^3 +108 g2^2 (Γ - κ2) + Sqrt[((Γ + κ1 - 2κ2)^3 + 108 g2^2 (-Γ + κ2))^2])^(1/3))/(12 2^(1/3))
Which is not 0. Clearly if o1 and o2 are common at that g1 value, subtracting them from one another should be zero. But clearly, that's not the case. I've also tried this for o2 and o3, and o1 and o3. But subtracting them from one another do not yield zero. What is the issue here?
Thanks!