  Let $P(x)\in \mathbb{Z}[x]$ be a polynomial with at least two distinct
complex roots. We prove that the number of solutions $(x_1, \dots, x_k, y_1,
\dots, y_k)\in [N]^{2k}$ to the equation \[
  \prod_{1\le i \le k} P(x_i) = \prod_{1\le j \le k} P(y_j)\neq 0 \] (for any
$k\ge 1$) is asymptotically $k!N^{k}$ as $N\to +\infty$. This solves a question
first proposed and studied by Najnudel. The result can also be interpreted as
saying that all even moments of random partial sums
$\frac{1}{\sqrt{N}}\sum_{n\le N}f(P(n))$ match standard complex Gaussian
moments as $N\to +\infty$, where $f$ is the Steinhaus random multiplicative
function.
