I have an equation in two variables $z$ and $\epsilon$. The equation is a sum of terms of the form $z^m\epsilon^n$ for $m,n$ integers (so both positive and negative exponents) which is equal to zero.
For a given $\epsilon$ there are several roots $z$. I have found that this equation is symmetric under the transformation $z\leftrightarrow z^{-1}$. How can I exploit this symmetry to make my life easier? I know that the roots will come in pairs $z_0$, $z_0^{-1}$, but is there anything else I can do? Often when there is some symmetry, you can make a change of variables to 'spend' this and make your life easier. Is there a way to do this here?
EDIT in response to YiFan's comment. I actually have a class of similar equations, which is why I am interested in general methods. In general I have $$\mathrm{det}\left\lvert\left(\begin{array}{cc}\alpha_0+\sum_{n=1}^N(z^n+z^{-n})\alpha_n-\epsilon & \sum_{n=1}^N(z^n-z^{-n})\beta_n \\ -\left[\sum_{n=1}^N(z^n-z^{-n})\beta_n\right] & -\left[\alpha_0+\sum_{n=1}^N(z^n+z^{-n})\alpha_n\right]-\epsilon\end{array}\right)\right\rvert=0,$$ where $\alpha_n,\beta_n$ are known real numbers, and $N$ is typically 1 or 2. I want to find all roots $z$ for a given $\epsilon$.
By standard theory of symmetric polynomials, if $f(x,y)$ is symmetric in $x,y$, then it is a polynomial function of the elementary symmetric polynomials, i.e., $f(x,y)=g(x+y,xy)$ for some polynomial $g$. In particular, $f(z,1/z)=g(z+1/z,1)$ will be a polynomial function of $w=z+1/z$. Therefore, instead of solving an equation of degree $2N$ and using the fact that roots come in pairs, you can solve an equation of degree $N$ to find $w$, and then solve $N$ quadratic equations to solve for $z$.