# Hyperbolicity of the partition Jensen polynomials

## TL;DR

This paper proves that partition Jensen polynomials become hyperbolic for large enough indices, providing explicit bounds for the minimal index where hyperbolicity occurs for degrees 3, 4, and 5.

## Contribution

It explicitly confirms the hyperbolicity of partition Jensen polynomials for all degrees beyond certain bounds, resolving a conjecture and providing concrete minimal index values.

## Key findings

- N(3)=94, N(4)=206, N(5)=381
- Explicit bounds for hyperbolicity onset for all degrees
- Confirmed hyperbolicity conjecture for partition Jensen polynomials

## Abstract

Given an arithmetic function $a: \mathbb{N} \rightarrow \mathbb{R}$, one can associate a naturally defined, doubly infinite family of Jensen polynomials. Recent work of Griffin, Ono, Rolen, and Zagier shows that for certain families of functions $a: \mathbb{N} \rightarrow \mathbb{R}$, the associated Jensen polynomials are eventually hyperbolic (i.e., eventually all of their roots are real). This work proves Chen, Jia, and Wang's conjecture that the partition Jensen polynomials are eventually hyperbolic as a special case. Here, we make this result explicit. Let $N(d)$ be the minimal number such that for all $n \geq N(d)$, the partition Jensen polynomial of degree $d$ and shift $n$ is hyperbolic. We prove that $N(3)=94$, $N(4)=206$, and $N(5)=381$, and in general, that $N(d) \leq (3d)^{24d} (50d)^{3d^{2}}$.