Paucity phenomena for polynomial products

Abstract

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.

8 pages; minor corrections; accepted version