Solution of a nonlinear equation

Question

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

Answer 1

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.

Answer 2

Assuming we are handling a polynomial identity and using a symbolic processor with

$$ \prod _{i=1}^{2 n} \left(\lambda -\lambda _i\right)-\left(\lambda \beta _{n-1}-\delta _{n-1}\right) \left(\prod _{i=2}^{n-1} \left(\lambda \alpha _i-\delta _i+\lambda ^2\right)\right)-\left(\lambda \alpha _n-\gamma _n+\lambda ^2\right) \left(\prod _{i=1}^{2 n-2} \left(\lambda -\mu _i\right)\right) = 0 $$

we can obtain the following relationships

$$ n = 1\to \left\{ \begin{array}{rcl} \delta _0+\gamma _1+\lambda _1 \lambda _2&=&0 \\ \alpha _1+\beta _0+\lambda _1+\lambda _2&=&0 \\ \end{array} \right. $$

$$ n=2\to\left\{ \begin{array}{rcl} \delta _1+\lambda _1 \lambda _2 \lambda _3 \lambda _4+\gamma _2 \mu _1 \mu _2&=&0 \\ \beta _1+\lambda _2 \lambda _3 \lambda _4+\lambda _1 \left(\lambda _3 \lambda _4+\lambda _2 \left(\lambda _3+\lambda _4\right)\right)+\alpha _2 \mu _1 \mu _2+\gamma _2 \left(\mu _1+\mu _2\right)&=&0 \\ \gamma _2+\lambda _2 \lambda _3+\left(\lambda _2+\lambda _3\right) \lambda _4+\lambda _1 \left(\lambda _2+\lambda _3+\lambda _4\right)-\mu _1 \mu _2+\alpha _2 \left(\mu _1+\mu _2\right)&=&0 \\ -\alpha _2-\lambda _1-\lambda _2-\lambda _3-\lambda _4+\mu _1+\mu _2&=&0 \\ \end{array} \right. $$

etc.

Follows the MATHEMATICA script

p[n_] := Product[lambda - Subscript[lambda, i], {i, 1, 2 n}] - (lambda^2 + Subscript[alpha, n] lambda - Subscript[gamma, n]) Product[lambda - Subscript[mu, i], {i, 1, 2 n - 2}] + (lambda Subscript[beta, n - 1] - Subscript[delta, n - 1]) Product[lambda^2 + Subscript[alpha, i] lambda - Subscript[delta, i], {i, 2, n - 1}]
Thread[CoefficientList[p[1], lambda]==0] // FullSimplify
Thread[CoefficientList[p[2], lambda]==0] // FullSimplify

Now defining $\mathcal{P}(x,n,m)$ as the sum of all possible $m-$product combinations of the list $\{x_1,\cdots,x_n\}$

ex. $\mathcal{P}(\lambda,3,2) = \lambda_1\lambda_2+\lambda_1\lambda_2+\lambda_1\lambda_3$

Now we can make some generalizations. For $n = 5$ we have

$$ \begin{array}{rcl} -\alpha_5 & = &\mathcal{P}(\lambda,2n,1)-\mathcal{P}(\mu,2 n-2,1)\\ \gamma_5 & = & -\mathcal{P}(\lambda,2n,2)+\mathcal{P}(\mu,2 n-2,2)-\alpha_5 \mathcal{P}(\mu,2 n-2,1)\\ -\beta_4 & = & \mathcal{P}(\lambda,2n,3)-\mathcal{P}(\mu,2 n-2,3)+\alpha_5\mathcal{P}(\mu,2 n-2,2)+\gamma_5\mathcal{P}(\mu,2 n-2,1)\\ -\alpha _2 \beta _4-\alpha _3 \beta _4-\alpha _4 \beta _4+\delta _4& = & -\mathcal{P}(\lambda,2n,4)+\mathcal{P}(\mu,2 n-2,4)-\alpha_5\mathcal{P}(\mu,2 n-2,3)-\gamma_5\mathcal{P}(\mu,2 n-2,2)\\ \phi_4(\alpha.\beta,\delta) & = & \mathcal{P}(\lambda,2n,5)-\mathcal{P}(\mu,2 n-2,5)+\alpha_5\mathcal{P}(\mu,2 n-2,4)+\gamma_5 \mathcal{P}(\mu,2 n-2,3)\\ \phi_5(\alpha,\beta,\delta) & = & -\mathcal{P}(\lambda,2n,6)+\mathcal{P}(\mu,2 n-2,6)-\alpha_5\mathcal{P}(\mu,2 n-2,5)+\gamma_5 \mathcal{P}(\mu,2 n-2,4)\\ \phi_6(\alpha,\beta,\delta) & = & \mathcal{P}(\lambda,2n,7)-\mathcal{P}(\mu,2 n-2,7)+\alpha_5\mathcal{P}(\mu,2 n-2,6)+\gamma_5 \mathcal{P}(\mu,2 n-2,5)\\ \vdots & & \vdots\\ -\delta _2 \delta _3 \delta _4^2 & = & -\mathcal{P}(\lambda,2n,10)-\gamma_5\mathcal{P}(\mu,2 n-2,8) \end{array} $$