Solution of a nonlinear equation I have the following equation that I need to solve
$$
\prod_{i=1}^{2n}(\lambda-\lambda_i) = (\lambda^2+\lambda\alpha_n-\gamma_n)\prod_{i=1}^{2n-2}(\lambda-\mu_i)-(\lambda\beta_{n-1}-\delta_{n-1})\prod_{i=2}^{n-1}(\lambda^2+\lambda\alpha_i+\delta_i)
$$
The two sets $\{\lambda_1, ...,\lambda_{2n}\}, \{\mu_1, ..., \mu_{2n-2}\}$ are given and four sets $\{\alpha_1, ..., \alpha_n\}, \{\gamma_1, ..., \gamma_{n}\}, \{\beta_1, ..., \beta_{n-1}\}, \{\delta_1, ..., \delta_{n-1}\}$ are unknown. The $\lambda$ is a variable can take any value.
To solve this equation, I equate the coefficients of $\lambda^{2n-1}, \lambda^{2n-2}, \lambda^{2n-3}$ of both sides, this ends up to the following values
\begin{align*}
\alpha_n &= \sum_{i=1}^{2n-2}\mu_i - \sum_{i=1}^{2n}\lambda_i\\
\gamma_n&= \sum_{1\leq i<j\leq2n}\lambda_i\lambda_j+\alpha_n \sum_{i=1}^{2n-2}\mu_i - \sum_{1\leq i<j\leq2n-2}\mu_i\mu_j\\
\beta_{n-1}&=\sum_{1\leq i<j<k\leq2n}\lambda_i\lambda_j\lambda_k+\alpha_n \sum_{1\leq i<j\leq2n-2}\mu_i\mu_j-\sum_{1\leq i<j<k\leq2n-2}\mu_i\mu_j\mu_k
\end{align*}
Now for $j=1,2,...,2n-2$ I let $\lambda = \mu_j$, then
$$
\prod_{i=1}^{2n}(\mu_j-\lambda_i) = -(\mu_j\beta_{n-1}-\delta_{n-1})\prod_{i=2}^{n-1}(\mu_j^2+\mu_j\alpha_i+\delta_i)
$$
It is a big system of nonlinear equations with $2n-1$ unknowns. I have no clear and direct way to go on. I would be appreciated for any constructive comment or idea that leads to solutions.
Thanks in advance 
 A: Disclaimer: I don't have a full answer. Just some ideas.
You have already managed to get $\alpha_n$, $\gamma_n$ and $\beta_{n-1}$. So it is safe to count them as known parameters. Now let $\lambda=0$, then
$$\prod_{i=1}^{2n}\lambda_i+\gamma_n\prod_{i=1}^{2n-2}\mu_i=\delta_{n-1}\prod_{i=2}^{n-1}\delta_i=\delta_{n-1}^2\prod_{i=2}^{n-2}\delta_i$$
The left side of the above equation is known. Let's call it $p$.
In the main equation, let $\lambda=\lambda_j$ for $j=1,2,...,2n$ to get
$$(\lambda_j^2+\lambda_j\alpha_n-\gamma_n)\prod_{i=1}^{2n-2}(\lambda_j-\mu_i)=(\lambda_j\beta_{n-1}-\delta_{n-1})\prod_{i=2}^{n-1}(\lambda_j^2+\lambda_j\alpha_i+\delta_i)$$
The left sides of these $2n$ equations are also known. Let's refer to the LHS as $P(\lambda_j)$ as it is a function of $\lambda_j$. So we have a set of $2n+1$ equations:
$$\begin{align}
P(\lambda_j)&=(\lambda_j\beta_{n-1}-\delta_{n-1})\prod_{i=2}^{n-1}(\lambda_j^2+\lambda_j\alpha_i+\delta_i),\qquad j=1,...,2n\\
p&=\delta_{n-1}^2\prod_{i=2}^{n-2}\delta_i
\end{align}$$
By the way, I didn't get where $\{\gamma_1, ..., \gamma_{n-1}\}$ and $\{\beta_1, ..., \beta_{n-2}\}$ enter the equations. They seem to have no role in your problem.
