This is similar to a question I just asked but the method there does not work here.
Consider the joint distribution of iid Cauchy random variables
$$f(x|\theta) = \prod_{i=1}^n \frac{1}{1+(x_i-\theta)^2}$$
My goal is to show that if
$$\prod_{i=1}^n \frac{1+x_i^2}{1+(x_i-\theta)^2} = \prod_{i=1}^n \frac{1+y_i^2}{1+(y_i-\theta)^2}$$
for at least $2n+1$ values of $\theta$, then the vectors $x,y$ are permutations of one another.
I have successfully shown that the above implies
$$\prod_{i=1}^n (1+(x_i-\theta)^2) = \prod_{i=1}^n (1+(y_i-\theta)^2)$$
for all values of $\theta \in \mathbb{R}$.
But here I am stuck. Essentially, I guess, I need to show that this factorisation is unique (up to permutations), but since this is not in the form $\prod_i (\eta_i-\theta)$ I cannot use the uniqueness of roots to complete this.
I am wondering if there is a theorem for polynomial factorisations that can help me finish this last step?
I know it's true as this is a well known result of the Cauchy distribution (despite the fact the proof is surprisingly hard to find...)
Edit
I just realised the above factorisation implies each polynomial has $2n$ complex roots of the form
$$x_i \pm i$$
$$y_i \pm i$$
I am unfamilar with complex roots, but I think this implies $x$ is a permutation has $y$, as a polynomial of degree $2n$ can have at most $2n$ roots, complex or real. Is this correct?