Partition Polynomials: Asymptotics and Zeros

TL;DR

This paper investigates the asymptotic behavior and zero distribution of partition polynomials, revealing complex multi-scale asymptotics and a network of zeros inside the unit disk, based on extensive computational experiments.

Contribution

It provides new insights into the asymptotics and zero distribution of partition polynomials, including the discovery of multi-scale asymptotics and their connection to dilogarithm curves.

Findings

Asymptotics of $F_n(x)$ have two scales: $n$ and $\sqrt{n}$.

Zeros of $F_n(x)$ form a network of curves inside the unit disk.

Computational experiments up to degree 70,000 support theoretical findings.

Abstract

Let $F_n(x)$ be the partition polynomial $\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of $F_n(x)$ have two scales of orders $n$ and $\sqrt{n}$ and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm.