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Let $P(x)\in \mathbb{Z}[x]$ be a polynomial with at least two distinctcomplex 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 questionfirst proposed and studied by Najnudel. The result can also be interpreted assaying that all even moments of random partial sums$\frac{1}{\sqrt{N}}\sum_{n\le N}f(P(n))$ match standard complex Gaussianmoments as $N\to +\infty$, where $f$ is the Steinhaus random multiplicativefunction.

Paucity phenomena for polynomial products