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Let $a,b,c$ and $t$ be complex numbers. How can one solve the following three equations for $a,b,$ and $c$ in terms of $t$:

$$ a^2 + b^2 + c^2 + 2 t =0, $$ $$ a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - 2 b^2 c^2 = 16, $$ and $$\Re\left(\int^b_a \sqrt{(z^2-a^2)(z^2-b^2)(z^2-c^2)} \ \mathrm{d}z \right)=0.$$ My attepmts using Solve, and NSolve have failed (even for a fixed $\tau$). Any help is very much appreciated.

My Attempts

K[a_, b_, c_] := Re[Integrate[Sqrt[(s^2 - a^2) (s^2 - b^2) (s^2 - c^2)], {s, a, b}]]
Solve[{a^2 + b^2 + c^2 + 2 t == 0, a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - 2 c^2 b^2 == 16, K[a, b, c] == 0}, {a, b, c}]

But it does not yield any ansswer. I get the error

Solve::nsmet: This system cannot be solved with the methods available to Solve.

These equations should have a solution, in particular, for any $t=\mathrm{i} y$, with $y>\sqrt{12}$. So I tried it with $y=4$

NSolve[{a^2 + b^2 + c^2 + 2 (4 I) == 0, 
  a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 - 2 c^2 b^2 == 16, 
  K[a, b, c] == 0}, {a, b, c}]

But I get the error

NSolve::nsmet: This system cannot be solved with the methods available to NSolve.
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    $\begingroup$ It would be more helpful if you a) showed your euqations as MMA code, not Tex, so we don't have to type them in again; b) showed the exact code you tried so far. $\endgroup$
    – MarcoB
    Dec 17 '20 at 19:06
  • $\begingroup$ @MarcoB Thanks for your comment. I edited the post. $\endgroup$
    – the8thone
    Dec 17 '20 at 19:36
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First, we specify t. Second, we reduce the system under consideration to the reals by the change a=a1+I*a2,b=b1+I*b2,c=c1+I*c2 and extracting the real and imaginary parts. Third, in order to solve that system in 6 variables, we find the minimum of the sum of the squared LHSes.

t = -5; NMinimize[Re[NIntegrate[
Sqrt[(z^2 - (a1 + I*a2)^2)*(z^2 - (b1 + I*b2)^2)*(z^2 - (c1 + 
I*c2)^2)], {z, a1 + I*a2, b1 + I*b2}]]^2 + 
ComplexExpand[Re[(a1 + I*b2)^2 + (b1 + I*b2)^2 + (c1 + I*c2)^2 + 2*t]]^2 + 
ComplexExpand[Im[(a1 + I*b2)^2 + (b1 + I*b2)^2 + (c1 + I*c2)^2 + 
2*t]]^2 + (ComplexExpand[Re[(a1 + I*b2)^4 + (b1 + I*b2)^4 + (c1 + I*c2)^4 - 
2*(a1 + I*b2)^2*(b1 + I*b2)^2 - 2*(a1 + I*b2)^2*(c1 + I*c2)^2 - 
2*(b1 + I*b2)^2*(c1 + I*c2)^2]] - 16)^2 + 
ComplexExpand[Im[(a1 + I*b2)^4 + (b1 + I*b2)^4 + (c1 + I*c2)^4 - 
2*(a1 + I*b2)^2*(b1 + I*b2)^2 - 2*(a1 + I*b2)^2*(c1 + I*c2)^2 - 
2*(b1 + I*b2)^2*(c1 + I*c2)^2]]^2, {a1, b1, c1, a2, b2, c2}, 
Method -> "DifferentialEvolution"]

{1.17036*10^-15, {a1 -> 2.64435, b1 -> 1.73702, c1 -> 2.79462*10^-9, a2 -> 5.46634*10^-10, b2 -> 7.91073*10^-10, c2 -> 0.0991094}}

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  • $\begingroup$ For t=4*I the usage of the options Method -> {"DifferentialEvolution", "ScalingFactor" -> 1}, AccuracyGoal -> 4, PrecisionGoal -> 5] gives {4.54936*10^-14,{a1->1.388,b1->1.17528,c1->-0.839586,a2->1.00926,b2->-1.30061,c2->0.793434}}. $\endgroup$
    – user64494
    Dec 17 '20 at 21:09

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