Paucity phenomena for polynomial products

TL;DR

This paper proves that the number of solutions to a polynomial product equation asymptotically matches a factorial times a power of N, confirming a conjecture and linking to Gaussian moments of random sums.

Contribution

It establishes the asymptotic count of solutions for polynomial product equations with at least two roots, solving a question posed by Najnudel.

Findings

Number of solutions asymptotically equals k! N^k

All even moments of certain random sums match Gaussian moments

Results confirm conjecture about polynomial product solutions

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.