Polynomial Partition Asymptotics

Abstract

Let \(f \in \mathbb{Z}[y]\) be a polynomial such that \(f(\mathbb{N}) \subseteq \mathbb{N}\), and let \(p_{\mathcal{A}_{f}}(n)\) denote number of partitions of \(n\) whose parts lie in the set \(\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}\). Under hypotheses on the roots of \(f-f(0)\), we use the Hardy--Littlewood circle method, a polylogarithm identity, and the Matsumoto--Weng zeta function to derive asymptotic formulae for \(p_{\mathcal{A}_f}(n)\), and the growth function \(p_{A_f}(n+1)-p_{\mathcal{A}_f}(n)\), as \(n\) tends to infinity. This generalises the analogous asymptotic formulae for number of partitions into perfect \(k\)th powers, established by Vaughan for \(k=2\), and Gafni for the case \(k \geq 2\), in 2015 and 2016 respectively.