Complex conjugate variables as independent variables in polynomial equations

[dm368]

Hi,
Is there any trick to treat complex conjugate variables in polynomial equations as independent variables by adding some other constraint equation ? Say, we have polynomial equation $f(x,x^{*},y,...) = 0$. where x^{*} is the complex conjugate of variable $x$. I might think of taking $x = r e^{i \alpha} =z_{1}$ and $x^{*} = r e^{-i \alpha} = z_{2}$ i.e. in polar form and then taking the original equation $f(z_{1},z_{2},y,...) = 0$ intersecting with $z_{1} z_{2} = 1$. But I don't know if this is the correct way - I am missing something here, right ?

Thanks in advance,

Cheers,

dm368

 

[fresh_42]

This depends on what you want to achieve. Beside the polar form you mentioned, you can always introduce the variables $\frac{1}{2}(x+x^*)\, , \,\frac{1}{2}(x-x^*)$ which are purely real and purely imaginary.