Solving a simple (but with multiplication between variables) system of 3 equations Here is a system of 3 equations with 3 unknowns $a$, $b$ and $c$.
$\begin{cases}
    a^2+\sigma \cdot a\cdot b + \sigma \cdot a \cdot c = n_1 \\
    b^2+\sigma \cdot a\cdot b + \sigma \cdot b \cdot c = n_2 \\
    c^2+\sigma \cdot a\cdot c + \sigma \cdot b \cdot c = n_3,
\end{cases}$
where $0<\sigma<1$, $n_1,n_2,n_3>0$ are all known.
This system seems quite simple but I cannot think of any method to solve it.
Q1: Is there a method to check the condition of this system to have real solution for $a,b,c$?
Q2: Is there a method to solve it analytically? Or at least to approximate it numerically?
 A: This is a very difficult problem and do not expect any analytical solution.
What I would do is to consider the three equations
$$f_1= a^2+\sigma \, a\, b + \sigma \, a \, c - n_1 $$
$$f_2= b^2+\sigma \, a\, b + \sigma \, b \, c - n_2 $$
$$f_3= c^2+\sigma \, a\, c + \sigma \, b \, c - n_3 $$ and define, as a norm,
$$\Phi(a,b,c)=f_1^2+f_2^2+f_3^2$$ and try to minimize it. If, at the solution, $\Phi=0$, you will have one solution.
For a test, I used $\sigma=0.678$, $n_1=18$, $n_2=28$, $n_3=39$ and the optimizer worked like a charm, leading to $\Phi=1.34\times 10^{-29}$ and
$$a=2.31796 \qquad \qquad b=3.44477\qquad \qquad c=4.58986$$ The initial data were generated using $a=2.34$, $b=3.45$, $c=4.56$ and the computed $n_i$ were rounded to the next integer.
In a second run, I added $15$, $10$ and $5$ to the $n_i$; no problem.
For me, this would be the safest approach.
For sure, it would be better to provide initial estimates for $(a,b,c)$. Since there are only three parameters, a preliminary grid search could be a good idea.
