Firstly, I have proved [Gamma] = [Beta]^2/12 is one of the solutions of (A12 P312 + A13 P213 + A23 P123)/P123123 + A12 A13 A23==0
In[1]:= \[Gamma] = \[Beta]^2/12;
l1 = 1/6 (Sqrt[\[Beta]^2 \[Omega]1^2 - 12 \[Gamma] \[Omega]1^2 -
12 k1^4 - 12 k1 \[Omega]1] - \[Beta] \[Omega]1);
l2 = 1/6 (Sqrt[\[Beta]^2 \[Omega]2^2 - 12 \[Gamma] \[Omega]2^2 -
12 k2^4 - 12 k2 \[Omega]2] - \[Beta] \[Omega]2);
l3 = 1/6 (Sqrt[\[Beta]^2 \[Omega]3^2 - 12 \[Gamma] \[Omega]3^2 -
12 k3^4 - 12 k3 \[Omega]3] - \[Beta] \[Omega]3);
A12 = (2 \[Gamma] \[Omega]1 \[Omega]2 + k2 (4 k1^3 + \[Omega]1) -
6 k1^2 k2^2 +
k1 (4 k2^3 + \[Omega]2) + \[Beta] (l1 \[Omega]2 + l2 \[Omega]1) +
6 l1 l2)/(
2 \[Gamma] \[Omega]1 \[Omega]2 + k2 (4 k1^3 + \[Omega]1) +
6 k1^2 k2^2 +
k1 (4 k2^3 + \[Omega]2) + \[Beta] (l1 \[Omega]2 + l2 \[Omega]1) +
6 l1 l2);
A13 = (2 \[Gamma] \[Omega]1 \[Omega]3 + k3 (4 k1^3 + \[Omega]1) -
6 k1^2 k3^2 +
k1 (4 k3^3 + \[Omega]3) + \[Beta] (l1 \[Omega]3 + l3 \[Omega]1) +
6 l1 l3)/(
2 \[Gamma] \[Omega]1 \[Omega]3 + k3 (4 k1^3 + \[Omega]1) +
6 k1^2 k3^2 +
k1 (4 k3^3 + \[Omega]3) + \[Beta] (l1 \[Omega]3 + l3 \[Omega]1) +
6 l1 l3);
A23 = (2 \[Gamma] \[Omega]2 \[Omega]3 + k3 (4 k2^3 + \[Omega]2) -
6 k2^2 k3^2 +
k2 (4 k3^3 + \[Omega]3) + \[Beta] (l2 \[Omega]3 + l3 \[Omega]2) +
6 l2 l3)/(
2 \[Gamma] \[Omega]2 \[Omega]3 + k3 (4 k2^3 + \[Omega]2) +
6 k2^2 k3^2 +
k2 (4 k3^3 + \[Omega]3) + \[Beta] (l2 \[Omega]3 + l3 \[Omega]2) +
6 l2 l3);
P312 = \[Gamma] (-\[Omega]1 - \[Omega]2 + \[Omega]3)^2 + (-k1 - k2 +
k3) (-\[Omega]1 - \[Omega]2 + \[Omega]3) + (-k1 - k2 +
k3)^4 + \[Beta] (-l1 - l2 +
l3) (-\[Omega]1 - \[Omega]2 + \[Omega]3) + 3 (-l1 - l2 + l3)^2;
P213 = \[Gamma] (-\[Omega]1 + \[Omega]2 - \[Omega]3)^2 + (-k1 + k2 -
k3) (-\[Omega]1 + \[Omega]2 - \[Omega]3) + (-k1 + k2 -
k3)^4 + \[Beta] (-l1 + l2 -
l3) (-\[Omega]1 + \[Omega]2 - \[Omega]3) + 3 (-l1 + l2 - l3)^2;
P123 = \[Gamma] (\[Omega]1 - \[Omega]2 - \[Omega]3)^2 + (k1 - k2 -
k3) (\[Omega]1 - \[Omega]2 - \[Omega]3) + (k1 - k2 -
k3)^4 + \[Beta] (l1 - l2 -
l3) (\[Omega]1 - \[Omega]2 - \[Omega]3) + 3 (l1 - l2 - l3)^2;
P123123 = \[Gamma] (\[Omega]1 + \[Omega]2 + \[Omega]3)^2 + (k1 + k2 +
k3) (\[Omega]1 + \[Omega]2 + \[Omega]3) + (k1 + k2 +
k3)^4 + \[Beta] (l1 + l2 +
l3) (\[Omega]1 + \[Omega]2 + \[Omega]3) + 3 (l1 + l2 + l3)^2;
Simplify[(A12 P312 + A13 P213 + A23 P123)/P123123 + A12 A13 A23 == 0]
Out[12]= True
Next, I want to solve the relation between [Beta] and [Gamma] through ``Reduce". However, it took too much time, and failed to give the output in finite time. How can I get the relation between [Beta] and [Gamma]?
ClearAll[\[Beta], \[Gamma]]
l1 = 1/6 (Sqrt[\[Beta]^2 \[Omega]1^2 - 12 \[Gamma] \[Omega]1^2 -
12 k1^4 - 12 k1 \[Omega]1] - \[Beta] \[Omega]1);
l2 = 1/6 (Sqrt[\[Beta]^2 \[Omega]2^2 - 12 \[Gamma] \[Omega]2^2 -
12 k2^4 - 12 k2 \[Omega]2] - \[Beta] \[Omega]2);
l3 = 1/6 (Sqrt[\[Beta]^2 \[Omega]3^2 - 12 \[Gamma] \[Omega]3^2 -
12 k3^4 - 12 k3 \[Omega]3] - \[Beta] \[Omega]3);
A12 = (2 \[Gamma] \[Omega]1 \[Omega]2 + k2 (4 k1^3 + \[Omega]1) -
6 k1^2 k2^2 +
k1 (4 k2^3 + \[Omega]2) + \[Beta] (l1 \[Omega]2 + l2 \[Omega]1) +
6 l1 l2)/(
2 \[Gamma] \[Omega]1 \[Omega]2 + k2 (4 k1^3 + \[Omega]1) +
6 k1^2 k2^2 +
k1 (4 k2^3 + \[Omega]2) + \[Beta] (l1 \[Omega]2 + l2 \[Omega]1) +
6 l1 l2);
A13 = (2 \[Gamma] \[Omega]1 \[Omega]3 + k3 (4 k1^3 + \[Omega]1) -
6 k1^2 k3^2 +
k1 (4 k3^3 + \[Omega]3) + \[Beta] (l1 \[Omega]3 + l3 \[Omega]1) +
6 l1 l3)/(
2 \[Gamma] \[Omega]1 \[Omega]3 + k3 (4 k1^3 + \[Omega]1) +
6 k1^2 k3^2 +
k1 (4 k3^3 + \[Omega]3) + \[Beta] (l1 \[Omega]3 + l3 \[Omega]1) +
6 l1 l3);
A23 = (2 \[Gamma] \[Omega]2 \[Omega]3 + k3 (4 k2^3 + \[Omega]2) -
6 k2^2 k3^2 +
k2 (4 k3^3 + \[Omega]3) + \[Beta] (l2 \[Omega]3 + l3 \[Omega]2) +
6 l2 l3)/(
2 \[Gamma] \[Omega]2 \[Omega]3 + k3 (4 k2^3 + \[Omega]2) +
6 k2^2 k3^2 +
k2 (4 k3^3 + \[Omega]3) + \[Beta] (l2 \[Omega]3 + l3 \[Omega]2) +
6 l2 l3);
P312 = \[Gamma] (-\[Omega]1 - \[Omega]2 + \[Omega]3)^2 + (-k1 - k2 +
k3) (-\[Omega]1 - \[Omega]2 + \[Omega]3) + (-k1 - k2 +
k3)^4 + \[Beta] (-l1 - l2 +
l3) (-\[Omega]1 - \[Omega]2 + \[Omega]3) + 3 (-l1 - l2 + l3)^2;
P213 = \[Gamma] (-\[Omega]1 + \[Omega]2 - \[Omega]3)^2 + (-k1 + k2 -
k3) (-\[Omega]1 + \[Omega]2 - \[Omega]3) + (-k1 + k2 -
k3)^4 + \[Beta] (-l1 + l2 -
l3) (-\[Omega]1 + \[Omega]2 - \[Omega]3) + 3 (-l1 + l2 - l3)^2;
P123 = \[Gamma] (\[Omega]1 - \[Omega]2 - \[Omega]3)^2 + (k1 - k2 -
k3) (\[Omega]1 - \[Omega]2 - \[Omega]3) + (k1 - k2 -
k3)^4 + \[Beta] (l1 - l2 -
l3) (\[Omega]1 - \[Omega]2 - \[Omega]3) + 3 (l1 - l2 - l3)^2;
P123123 = \[Gamma] (\[Omega]1 + \[Omega]2 + \[Omega]3)^2 + (k1 + k2 +
k3) (\[Omega]1 + \[Omega]2 + \[Omega]3) + (k1 + k2 +
k3)^4 + \[Beta] (l1 + l2 +
l3) (\[Omega]1 + \[Omega]2 + \[Omega]3) + 3 (l1 + l2 + l3)^2;
Reduce[(A12 P312 + A13 P213 + A23 P123)/P123123 + A12 A13 A23 ==
0, {\[Beta], \[Gamma]}, Reals]
This question is also posted at https://mathematica.stackexchange.com/questions/289814/symbolic-computation-using-reduce-does-not-give-a-result
Attachments:
