Let $c_E$, $c_F$, $c_G$ and $c_H$ be known real numbers.
Let $x_1$, $w_1$, $x_2$ and $w_2$ be unknown real numbers.
When solving a cubic equation, one could be led to solve that kind of system:
\begin{equation} \begin{split} (x_1+\text{i}w_1)+(x_2+\text{i}w_2)&=8(c_G+\text{i}c_H)\\ \sqrt[3]{x_1+\text{i}w_1}\sqrt[3]{x_2+\text{i}w_2}&=4(c_E+\text{i}c_F)\\ \end{split} \end{equation}
Raising the second equation to the cube will allow to find $x_1$, $w_1$, $x_2$ and $w_2$ by solving a quadratic equation.
However it seems to also result in a loss of information about the relation between the arguments of $\sqrt[3]{x_1+\text{i}w_1}$ and $\sqrt[3]{x_2+\text{i}w_2}$, which appear in the expressions of the cubic's roots.
This problem has been underlined at Cubic formula gives the wrong result (triple checked) .
1) If $(c_E,c_F)=(0,0)$ then at least one of the complex numbers $(x_j+\text{i}w_j)$ is $0$, for $j\in\{1,2\}$.
Then there is no constraint on the argument of the other one. Is it correct?
2) If $(c_E,c_F)\neq(0,0)$ then I guess there is a degree of freedom allowing to choose the argument of one complex number, and then the other is determined by the second equation, isn'it?
A way to take this into account has been provided at Why not write the solutions of a cubic this way? , but what if one wants to have no cube root in the denominator?
2.1) ATTEMPT 1:
I simply rewrote the second equation as:
\begin{equation} \sqrt[3]{x_2+\text{i}w_2}=\frac{4(c_E+\text{i}c_F)\left(\sqrt[3]{x_1+\text{i}w_1}\right)^2}{x_1+\text{i}w_1} \end{equation}
It seems to always work. However, is there a more elegant way to write the square of the cube root ?
(I know that $\frac{4(c_E+\text{i}c_F)\sqrt[3]{(x_1+\text{i}w_1)^2}}{x_1+\text{i}w_1}$ will always work for real numbers by choosing the real cube root, but it won't always work for other choices or unreal numbers).
2.2) ATTEMPT 2:
$\text{atan2}(x_1,w_1)$, $\text{atan2}(x_2,w_2)$ and $\text{atan2}(c_E,c_F)$ are defined.
Then, for each $j\in\{1,2\}$ and $k_j\in\{-1,0,1\}$, I can write:
\begin{equation} \sqrt[3]{x_j+\text{i}w_j}=\sqrt[6]{x_j^2+w_j^2}\exp\left(\frac{\text{i}}{3}\text{atan2}(x_j,w_j)+\frac{2k_j\pi\text{i}}{3}\right) \end{equation}
For any real integer $k_{EF}$, I can also write:
\begin{equation} c_E+\text{i}c_F=\sqrt{c_E^2+c_F^2}\exp\left(\text{i}\ \text{atan2}(c_E,c_F)+2k_{EF}\pi\text{i}\right) \end{equation}
I included these equations in the second equation of the initial system:
\begin{equation} \begin{split} &\sqrt[6]{x_1^2+w_1^2}\sqrt[6]{x_2^2+w_2^2}\exp\left(\frac{\text{i}}{3}\text{atan2}(x_1,w_1)+\frac{\text{i}}{3}\text{atan2}(x_2,w_2)+\frac{2(k_1+k_2)\pi\text{i}}{3}\right)\\&=4\sqrt{c_E^2+c_F^2}\exp\left(\text{i}\ \text{atan2}(c_E,c_F)+2k_{EF}\pi\text{i}\right) \end{split} \end{equation}
As far as I know, both norms in this last equation are necessarily equal, thus both arguments are equal as well.
Then I can write the argument of $(x_2+\text{i}w_2)$ as:
\begin{equation} \begin{split} &\frac{\text{i}}{3}\text{atan2}(x_2,w_2)+\frac{2k_2\pi\text{i}}{3}=\frac{\text{i}}{3}(3\ \text{atan2}(c_E,c_F)-\text{atan2}(x_1,w_1))+\frac{2(3k_{EF}-k_1)\pi\text{i}}{3}\\ &=\frac{\text{i}}{3}\text{atan2}(c_E(c_E^2-3c_F^2)x_1+c_F(3c_E^2-c_F^2)w_1,c_F(3c_E^2-c_F^2)x_1-c_E(c_E^2-3c_F^2)w_1)+\frac{2(3k_{EF}-k_1)\pi\text{i}}{3}\\ \end{split} \end{equation}
The value of $k_{EF}$ doesn't seem to change anything and can always be chosen as 0, right?
The right-hand-side of the first line seems to always provide the good argument.
However the right-hand-side of the second line, obtained by applying the formula for the sum of atan2, seems to have again resulted in a loss of information.
What's wrong with it?
2.3) AESTHETICS
Whathever the method is, it always seems to favour one complex number over the other.
Is there a more "neutral" way to get an always working expression?
Thanks in advance for your answers.
